lord kelvin volume 4

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i c D o m a i n

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s s_ u s e # p d

Mathematical and physical papers, by Sir William Thomson. Collected

from different scientific periodicals from May, 1841, to the present

time.

Kelvin, William Thomson, Baron, 1824-1907.

Cambridge, University Press, 1882-1911.

http://hdl.handle.net/2027/miun.aat1571.0004.001

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R SITYPR ESS

NE E. C .

ER

R INC ESSTREET

.

C H AU S

N A M S S O N S

a lcutta : MA C MILLA NA NDCO . LTD.

P u b l i c D o m a i n


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GENERALDYNAMICS

R A B L E

N B A R O N K E L V I N

O . L L .D . D . C . L. S C .D . M . D . . . .

R . A S S O C . I NS T IT U T E O F F R A NC E

T H E L E GI O N O F H O N O U R K T P RU S S I A N O R D ER P O U R L E M IA R IT E

H E U N I V E R SI T Y O F G L AS G O W

E TE R S C O L L E GE C A MB R I D G E

EDWITHB R IE A NNOTA TIONSB Y

R D . Sc . L L .D . S E C. R .S .

SOR O F MATHEMA TIC SINTHEUNIV ER SITYO F CA MB R IDGE

S T J O H N S C O L LE GE

R ESS



s s_ u s e # p d



L A Y M . A.

R ESS



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esofthisreprintthe paperswere

y f romItoC IV thosewhichhad

hev o lumeof " PapersonElectrostaticsandMagnetism ( Macmillans 1872 reprinted1884 ,

withareferencetotheir placein

inhimselfbeganthepreparationof

dmateria lw hichhadbeenstandingintype

sultimatelyprintedoff asan

9 , to thevo lumeof " B a lt imoreLectures

henumberingofthesepapersi sin

nsafterwardsmadeto V olumeIII.

hev o lumeof " B a lt imoreLectures

itherinthete torasA ppendices aconsiderablenumberof la terpapersconnectedwiththeDynamica l

erattheend ofV olumeIII.ofthe

sicalPapers ( 1890 he inserteda

nly j ustw ritten onthere lations

amicpropagation stimulatedthereto

natingcurrentsin cables andof

esinspace boththenundergo inge ploration. A lsoaconsiderablenumberof the lessabstract

edandreprintedwithoutregardto date

PopularLecturesandA ddresses

C onstitutionofMatter 1889 V o l. II.

sics 1894 V o l. III. Na igationa l

w henhewasre uestedbyLady

geof thecompletionof theco llectededit ion

o r , t h at i n c on s e u e nc e o f th e v a r i e ty o f

eprints anyattemptatcontinuing



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ofthepapersmustbe abandoned.It

thataclearerv iew of these uenceofLord

acti ity couldbeobta inedbyclassif y ingthe

numberofbroadheadings co llecting

nchronologicalorder thematerial

datthesametimema ingtherecord

ludingtitlesof otherpaperswith

heretheyhad alreadybeenrepublished.Thisprocedurehasbeencarriedbac intimefar

nne ionw ithLordK el in sow n

isearlierw or asreprintedinthe

thisco llection. Inordertosecurethe

ntinuityundervariousheadings w here

d ithasbeenthoughtad isable

adyincludedin the" B altimore

re .

edinthis rearrangementofthe

y theuseof tw o importantbibliographiesofLordK el in sw or . F orthepapersupto1884

of theR oyalSociety sC ata logueof

a ilable andfortheremainingperiod

hroughthecourtesyofP ro f . McLeod to

the continuationofthatcatalogue.

presentv o lumeandallthene t

graphyof661titlesappendedto

n sLifeofLordKel inhask indlybeen

amepurpose.Inthat listthecrossreferencestoreprintsor abstractsofthev ariouspapersare

suchisthecomple ityof themateria l

hhasbeenre uiredtoestablishthe

ariousentries.Aconsiderableproportionofthe listconsistsoftitles ofv erbalcommunications

ng madetolearnedSocieties where

yanaccidenta lpressabstract hasbeen

specti tisofcourse morecomplete

publicationswhichaloneisgi en

thetableofContents.



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pp.270-456 onW a esonW ater.

ysicalside asub ectpre-eminently

mathematicalmachinerywasformulated

-perhapsmainlyow ingtothere uirementsof thescientif icengineersw hode eloped f irstthe

andaf terw ardsthedesigningofships

ngthedrainon thepropellingenergy

ductionofw a es. LordKel in s

ac uiredasayachtsman ledhimdirectly

ef fecto fw indandcurrent inw hich

LordR ay leigh and mainlyonthe

Helmholt w hile thesearchforthe

sistanceto ships andthee perimentsofO sborneReynoldsonthedemarcationbetweensmooth

mpteddiff icult in estigationsinv iscous

mplete.Themodein whichthe

arofalimitedregular trainof

t hasfeatureswhichare importanta lso

ead anceofabeamof radiation

um w hiletheregulartra insofstanding

acurrent by f low o erasubmerged

modeofgenesiso fw a ymotionw hichmay

eorologicalatmosphericphenomena.

tionisthegraphicalrepresentations

sareduetotheRoya lSocietyofEdinburgh

of thenumerousdiagrams.

eneralDynamics pp.457-5 1

ariousfragmentarypapers beginning

nofthepartitionofthermalmolecular

ntoapplicationsofthePrinciple ofAction

odstothesub ectofperiodicorbits

micalastronomy andtothegraphical

lems.Asfollowingnaturallyonthis

e iscompletedbyabrie f sectiononElastic

2-560 w hichislitt lemorethanachrono logica ll istof t it lesofpapers mainlyofoptica landelectrical

t ionandrefle ioninordinarye lasticsolid

les w hichha ebeenrepublished



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s. Thespecia lsub ecto fpropagation

sideredinitsmoremodernelectric

e n re s er e d f or t h e ne t v o l um e .

. oftheMathematicalandPhysical

s now arrangedready forpress on

micaland GeologicalPhysics ElectrodynamicsandElectrolysis MolecularandCrystallineTheory

onicTheory withperhapssomeaddressesandothermiscellaneousscientificmatter.

nowledgemuche pertassistance

dhistas .Manyoftheproofsheets

o ug h b y Mr W . J . H A RR I SO N F e l l o w of

ge.Inthecorrectionofthe latterhalf

gilanceofMrGEO R GEGR EEN w how as

tif icsecretary forthe lateryearsofhislif e

ecia lk now ledgetobear hasensured

mallo ersights inaddit iontomore

e plicitlymentionedinfootnotes.

. O R R F . R . S. w hoseassistancew asspecially

onw iththedif f iculttopicstreatedonp. 33 0

nproofa llthesubse uentsheetsof the

a lsosuppliedmostof the listo ferratabelongingtotheearlierpart. F orgenera lad icere latingto

theEditorisunderobligationtoLordR A YLEIGH

, andtoDrJ . T . B O TTOMLEY. Inthe

rerrorsandmisprintsha ebeencorrected

a ll importantchangesha ebeenreferred

he Editor whichareenclosedin

ways totheofficialsofthe Cambridge

hee ce llenceof theirw or , andtheir

E GE C A MB R I D G E.



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PAGE

oms. . .. . . .. 1

Motion. .. . . .. 1

tryV elocityofaC ircularV orte R ing. 67

nofF reeSo lidsthroughaLi uid. 69

indandC apillarityonWa esinw ater

76

es. . .. . . . 86

cese periencedbySolidsimmersedina

9

sandRepulsionsdueto V ibration. 98

tionofR igidSo lidsinaLi uidcirculating

orationsinthemorina

01

atics. . .. . . . 115

essiona lMotionofaLi uid[ Li uid

nets[ i l lustratingV orte -Systems . . 1 5

tationa lOscil la t ionsofR otatingWater. 141

mationofC ore lessV orticesby theMotion

iscidIncompressibleF luid 149

faC o lumnarV orte . . . . 152

ityofSteadyandofPeriodicF luidMotion 166

bngInf inity inLordR ay leigh sSo lution

orte Stratum.. . 186

ragePressuredueto impulseofV orte -R ings

uresofE uil ibriumofaR otatingMassof

89

nofaLi uidw ithinanEll ipso idalHo llow 19

ityandSmallOscil la t ionofaPerfect

ightC ore lessV ortices. 202



s s_ u s e # p d



PAGE

ff iciencyofSa ils Windmills ScrewPropellersinWaterandA ir andA eroplanes. 205

sistanceofaF luidtoaP lanek eptmo ing

inedto itata small

nofaHeterogeneousLi uid commencing

Motionof itsB oundary. 211

rneofDiscontinuityofF luidMotion in

anceagainstaSolid

id.. . . . 215

ES.

edErrorinLaplace sTheoryof theT ides. 2 1

O scil lat ionsof theF irstSpecies in

Tides.... 248

ationofLaplace sDif ferentialE uation

R .

ryWa esinF low ingWater. . . 270

esproducedbyaSingle Impulse inWater

spersi eMedium. . 30

rontandRearofaF reeProcessionofWa es

07

a e s .. . .. . .. 3 0 7

PropagationofLaminarMotionthrougha

iscidLi uid... 3 08

otionofV iscousF luidbetw eentwo

1

aterTw o-DimensionalWa esproducedby

urbance. 33 8

rontandRearofaF reeProcessionofWa es

51

rShip-Wa es. .. . . . 368

hip-Wa es. .. . . .. 3 94

Deep-SeaWa esofThreeC lasses: ( 1 f rom

2 f romaGroupofE ua l

s ( 3 ) byaPeriodica lly

re... 419

lanationof theMac ere lS y . . . 457



s s_ u s e # p d



PAGE

inematica landDynamicalTheorems. 458

ormofC entrifuga lGo ernor. . . 460

w A stronomica lC loc , andaPendulum

Motion. . .. 46

bationsoftheCompassproducedby the

64

orm ofA stronomica lC loc w ithF ree

ntlyGo ernedU niform

heel.... 470

wedasPossiblyaModeofMotion. 472

saK ineticTheoryofMatter... 474

ticW or ingModeloftheMagneticCompass.......... 475

periments...... 482

tC asesfortheMa w ell-B o lt mann

utionofEnergy.. 484

eTest-casedispro ingtheMa w ell-B o lt mannDoctrineregardingDistributionofKinetic

otionofaF initeC onser ati eSystem. 497

reminP laneKineticTrigonometrysuggested

C ur atura Integra. . 51

tyofPeriodic Motion.... 515

olutionof DynamicalProblems.. 516

e eryProblemofTw oF reedomsinC onser ati eDynamicstotheDraw ingofGeodetic

enSpecif icC ur ature. 521

onof " Mercator s P ro ectionperformedby

ts.... 52

catorChartononeSheetrepresenting

lyContinuousClosed

lProblems...... 5 1

nturyCloudso ertheDynamicalTheory

1



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N.

PAGE

InductionofElectric CurrentsinSubmarineTelegraphW ires..... 5 2

ngthroughSubmarineC ables i l lustratedby

ghaModelSubmarine

MirrorGa l anometerandby

2

ustrationsoftheMagneticandthe HelicoidalRotatoryEffectsofTransparentB odieson

2

ndWa esinaStretchedUniformC hain

.... 5 3

TheoryofLight.. . . .. 5 8

sandGreen sDoctrineofE traneousF orce

F resne l sK inematicsof

8

thesisforElectro-magneticInductionof

thconse uentE uationsof

dHomogeneousSo lidMatter 5 9

erenceofElectricitywithina HomogeneousSolidConductor..... 545

icationsofF ourier sLaw ofDif fusion i l lustratedbyaDiagramofC ur esw ithA bso lute

546

ghtningConductorsattheB ritish

ionandR ef ractionofLight. . . 547

ticity andPonderableMatter. .. 547

anismfortheC onstitutionofEther. . 547

iscousLi uid E uil ibriumorMotionof

l ibriumorMotionofan

re ity " Ether" ; Mechanica lRepresentationofMagneticF orce. . 547

perimentsforComparingtheDischarge

Dif ferentB ranchesofa

o r d K e l i n a nd A l e a n de r

aTheoryofR ef raction Dispersion and

. 551

ndulatoryTheoryofC ondensationa lrarefactionalW a esinGases Li uids andSo lids

nSo lids o fElectricWa es

f Transmittingthem

isibleLight U ltra - io let



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X V

PAGE

tionandRefractionofSolitaryPlane

rfacebetweenTw oIsotropic

So lid orEther. . 551

ellmeier sDynamicalTheorytothe

oducedbySodium- apour. 551

ationofF orcew ithinaLim itedSpace

herica lSo litaryWa es or

es ofbothSpecies E ui oluminalandIrrotational inanElasticSolid. 552

producedin anInfiniteElasticSolidby

ceoccupiedby itofa

yAttractionorRepulsion. 552

of EtherforElectricityandMagnetism. 55

yingMethodforStress andStraininan

56

ctro -etherealTheoryof theV elocityofLight

a n d So l id s .. 5 6 0



s s_ u s e # p d



f oo t f o r m om e nt r e ad m o me n tu m , 7 , C f . L a mb s H y dr o dy n am i cs ~ ~ 1 2 9 1 0 , 1 0 2 f i rs t e . f o r r re a d T , 1 6 l i ne 8 f r om f o ot f o r di m in i sh i ng r e ad i n cr e as i ng , 1 6 , , 2 , f o r - re a d + , , 1 4 1 , , 3 , f o r pp . 9 7- 1 09 r e ad p p . 10 9 -1 1 6

, e . ( 8 ) , f o r - - d2 r ea d2 d -2 , 1 4 , l i n e1 0 fr om f oo t o m it i n af te r U s in g , 1 4 4 , , 5 f or 0 r ea d a , 1 6 , , 8 f or i = l r ea d i= 0 , 1 6 0 , , 1 2 f or 4 r ea d 2 , 1 65 f oo tn ot e r e ad [ s up ra p . 1 , , 1 7 5 l i n e7 o mi t e e r y .C f. L am b s H yd ro dy na mi cs ~ 1 64 , 1 86 s ee f oo tn ot es p . 3 3 4 , 2 5 5 e . ( 3 ) a nd ( 4 , d e le te r , 2 5 5 , , ( 6 , f o r r ea d r2 2 56 , , ( 8 , ( 9 , ( 1 1 , ( 1 2 , f o r r re ad r 2 , 3 0 5 , , ( 1 4 , f o r 22 i n la st e p re ss io n re ad 2 , 3 1 7 se . Re fe re nc e ma y be m ad e to W . M .H ic s B r i t .A ss oc . Re po rt 1 88 5

p. 517 a lsop. 9 0: a lsotosameauthor Phil.

8 , p.3 3 , on" SpiralorGyrostaticV orte



s s_ u s e # p d



T M S .

ya lSocietyofEdinburgh V o l. v I pp. 94-105

o l . x x x i 1 8 67 p p . 15 - 24 .

ot ' sadmirabledisco eryof the law of

rfectli u id- thatis inaf luidperfectly

orf luidf rict ion - theauthorsa idthatthis

suggeststhe ideathatHelmholt ' sringsare

rtheonlyprete tseemingto j ustif y

nofinfinitelystrongandinfinitelyrigid

ee istenceofw hichisassertedasaprobable

greatestmodernchemistsin their

rystatements isthaturgedbyLucretius

hatitseemsnecessarytoaccountfor

hingq ualitiesofdifferentk indsof

haspro edanabsolute lyuna lterable

anyportionofaperfectli u idinw hich

heca lls" Wirbe lbewegung hasbeen

rtionofa perfectli uidwhichhas

hasonerecommendationofLucretius satoms

cq uality.Togenerateortodestroy

naperfectf luidcanonlybeanactofcreati e

mdoesnote pla inanyof theproperties

1



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ngthemtothe atomitself.Thusthe

a s i t ha s b ee n w el l c al l ed h a s be e n in o e d b y

countforthe elasticityofgases.E ery

assimilarlyre uiredanassumption

gtothe atom.Itisas easy( andas

moreso toassumew hate erspecif icforces

ortionof matterwhichpossessesthe

asinaso lidindi isiblepieceofmatter and

hasnoprimafaciead antageo erthe

nificentdisplayofsmo e- rings w hich

reof witnessinginProfessorTait s

edbyonethenumberofassumptionsre uiredtoe plainthepropertiesofmatteron thehypothesisthat

orte atomsinaperfecthomogeneous

ngsw eref re uently seentoboundobli ue ly

ingv io lently f romtheef fectsof theshoc .

lartothatobser ableintwolargeindiarubberrings stri ingoneanotherinthe air.Theelasticityof

dnofurtherfromperfectionthanmightbe

- rubberringof thesameshape f rom

scosityof india - rubber. O fcoursethis

isperfecte lasticity forv orte ringsina

astasgoodabeginningasthe" clashof

eelasticity ofgases.Probablythe

ofD. B ernoull i Herapath J oule K ronig

ll onthevariousthermodynamicproperties

lltheposit i eassumptionstheyha ebeen

stomutua lforcesbetw eentw oatomsand

edby indi idua latomsormolecules satisf ied

thoutre uiringanyotherproperty inthe

posesthemthaninertiaandincompressibleoccupationofspace.A fullmathematicalin estigation

weentw ovorte ringsofanygi en

espassingoneanotherinany twolines

ercomeneareroneanotherthana

meterofe ither isaperfectly so l able

andtheno eltyof thecircumstances

fficultiesofane citingcharacter.Its

undationoftheproposednew k inetic

bilityoffoundinga theoryofelastic



s s_ u s e # p d



M S

dynamicsofmoreclose ly-pac edv orte

anticipated.Itmayberemar edin

icipation thatthemeretit leo fR an ine s

ortices communicatedtotheRoya l

849and1850 w asamostsuggesti e

swereshowntothe Societyto

ittedv orte atoms theendlessvarietyof

ansuff icienttoe pla inthev arieties

nsimplebodiesandtheirmutualaffinities.

ttworingatomslin edtogetherorone

w ithitsendsmeeting constituteasystem

aybealteredinshape canne erde iatef rom

plecontinuity itbeingimpossiblefor

orte motiontogothroughthe lineof

motionoranyotherpartof itsownline.

orte core islitera lly indi isiblebyany

rte motion.

ntoa v eryimportantpropertyof

ithreferencetothenowcelebratedspectrumanalysispracticallyestablishedbythedisco eriesandlaboursof

. Thedynamica ltheoryof thissub ect

adtaughtto theauthorofthepresent

852 andwhichhehastaught inhis

sityofGlasgow f romthattimeforw ard

econstitutionofsimple bodiesshould

amentalperiodsofv ibration ashasa

ormorestrings oranelasticsolid

tuning-for srigidlyconnected.To

ntheLucretiusatom isatoncetogi e

andelasticity forthee planationofw hich

tebodies theatomicconstitutionw as

hen thehypothesiso fatomsandv acuum

hisfollowersto benecessarytoaccount

mpressibilityoftangiblesolidsand fluids

twouldbenecessarythatthe moleculeof

shouldbenotanatom butagroupofatoms

nthem.Suchamoleculecould notbe

dthusitlosestheonerecommendation

egreeofacceptance ithashadamong



s s_ u s e # p d



sthee perimentsshow ntotheSociety

atomhasperfectlydefinite fundamenta l

dependingsole lyonthatmotionthee istence

disco eryofthesefundamentalmodes

tingproblemofpuremathematics.

o lt ring theanaly ticaldif f icult ieswhich

formidablecharacter butcerta inly far

sentstateof mathematicalscience.

ommunicationhadnot attempted

ute ceptforaninf inite ly long stra ight

orthiscasehew asw or ingoutso lutions

possibledescriptionofinfinitesimalv ibration andintendedtoincludethemin amathematicalpaperwhich

ocommunicatetotheRoyalSociety.

whichhe couldnowstateisthe following.

enw ithitssectiondif feringf rome act

tesimalharmonicde iationoforderi.

a esroundthea isofthecylinder in

vorte rotation w ithanangularv e locity

heangularve locityof thisrotation. Hence as

w holecircumference ise ua lto i f o ran

rderithereare i-1periodsofv ibration

ionof thevorte . F orthecase i= 1

andthesolutione pressesmerelyaninf initesimallydisplacedv orte w ithitscircularformunchanged.

dstoellipticdeformationofthe circular

eriodofv ibrationis therefore simply

. Theseresultsare o fcourse applicable

henthediameterof theappro imately

comparisonwiththediameterofthe

o e- ringse hibitedtotheSociety . The

esof thetw ok indsof trans ersev ibrationsofaring suchasthev ibrationsthatw ereseeninthe

bemuchgra erthantheell ipticv ibrationof

thev ibrationswhichconstitutethe

apourareanalogoustothosewhichthe

bited andit isthereforeprobablethatthe

rotationof theatomsofsodium- apouris

illionthofthemillionthof asecond

ey theperiodofv ibrationof theye llow



s s_ u s e # p d



M S

inasmuchasthislightconsistso f tw o

istentinslightlydif ferentperiods e ua l

mej uststated andofasnearlyascan

tensit ies thesodiumatommustha etw o

bration ha ingthoseforthe irrespecti e

ute ua llye citablebysuchforcesasthe

ncandescentv apour.Thislastcondition

twofundamentalmodesconcerned

ar( andnotmere lydifferentordersof

oconcurv erynearlyintheirperiods of

pro imatelycircularanduniformdis o f

ntalmodesof trans ersev ibration w ith

adrants fulf ilboththecondit ions. Inan

anduniformringofe lasticso lidthesecondit ionsarefulfi l ledforthef le ura lv ibrationsinitsplane anda lso

tionsperpendicularto itsow nplane. B ut

g if createdw ithonepartsomew hatthic er

tremainso butw oulde perience longitudina lv ibrationsrounditsow ncircumference andcouldnot

mentalmodesofv ibrationsimilarin

atelye ua linperiod. Thesameassertion

bepractica llye tendedtoanyatomconsistingofasinglev orte ring how e erin o l ed asil lustrated

wnto theSocietywhichconsistedof

edinvariousw ays. Itseems therefore

tommaynotconsist ofasinglev orte

probablyconsisto f tw oappro imatelye ua l

hroughoneanotherli e tw o lin sofacha in.

uitecerta inthatav apourconsistingofsuch

o lumesandangularve locit iesinthetw o

uldactprecise lyasincandescentsodium apouracts- thatistosay w ouldfulf i lthe" spectrumtest for

geoftemperatureon thefundamentalmodescannotbepronounceduponwithoutmathematical

toe ecuted andthereforew ecannotsay

867. -Theauthorhasseenreasonforbelie ingthatthe

htbereali edbyacertain configurationofasingleline

bedescribedinthemathematica lpaperw hichheintendsto

ty.



s s_ u s e # p d



M S

isthroughthecentreof thering for

esidethe lineofmotionof - theringsees as

heposit ionofhiseye acon e 4outline

e inf ronto f thering. Thiscon e

dingsurfacebetweentheq uantityof

forw ardwiththeringinitsmotionand

ieldsto letitpass.It isnotsoeasy

ondingcon e outlinebehindthering

smo eisgenerallyleftin therear.In

gsurface oftheportioncarriedforward

itesymmetricalontheanteriorand

eplane ofthering.Themotionof

beprecisely thesameasitwould beif

e wereoccupiedbyasmoothsolid

itis inastateof rapidmotion

ara isofthering withincreasing

nearerandnearertothering itself.The

lmotionmaybe imaginedthus:-Leta

ber o fcircularsection w ithadiameter

ength bebentintoacircle anditstwo

thersothat itmayk eepthecircular

lettheapertureofthe ringbeclosedby

tanimpulsi epressurebeapplieda ll

ensity sodistributedastoproducethedef inite

cifiedasfollows andinstantlythereafter

ed. Thismotionis inaccordancewith

w s tobea longthosecur esw hichw ouldbe

inplaceof the india -rubbercircle w eresubstitutedaringe lectromagnetS andtheve locit iesatdif ferentpo ints

ntsprecise ly thecon e outlinereferredto andthe lines

luidcarrieda longby thev orte forthecaseofadouble

twoinf inite ly long para lle l stra ightv orticesofe ua lrotations

ecur esha ebeendraw nbyMrD. M F arlane f rom

erformedbymeansof thee uationofthesystemof

x + b

w h e r el o g N = a

a

nthemathematicalpaperwhichthe authorintendsto

eto theRoyalSocietyofEdinburgh.

arconductorwitha currentofelectricitymaintained



s s_ u s e # p d



heintensitiesofthe magneticforcesin

fthemagneticfield.The motion as

w illf ulf i lthisdef inition andw illcontinue

ingv e locit iesate erypo into f thef ilm

anebe inproportiontotheintensities

ecorrespondingpointsofthemagnetic

mo edperpendiculartoitsown plane

otionofthefluidthroughthe middle

ocityverysmallincomparisonw iththat

f thering. A largeappro imately

willbecarriedforwardwiththe ring.

ringbeincreased thevo lumeof f luid

minishedine erydiameter butmostin

tdiameter anditsshapew illthusbecome

asingthev elocityofthering forward

blatenessw ill increase until insteadofbe ing

il lbeconca ebeforeandbehind roundthe

theforw ardve locityof theringbe

te ua ltotheve locityof thefluidthrough

hea ia lsectionof theoutlineof the

ardwill becomealemniscate.Ifthe

orward theportionofit carriedwith

be itselfannular and re lati e ly tothe

f luidw illbebac w ardsthroughthecentre.

eportionof fluidcarriedforwardand

symmetrica l bothre lati e ly tothe

thetwosidesof thee uatoria lplane. A ny

nthusdescribedmightofcoursebe

derdescribed orby f irstgi ingav e locity

gthefluidin motionbyaidofan

byapply ingthetw oinit ia ti eactions

amountof the impulsere uired or

f fecti emomentumof themotion or

hemotion isthesumof theintegral

ontheringandonthefilmre uiredto

etwocomponentsofthewholemotion.

sthediameterof theringisv erysmall

ameterof thecirculara is the impulse

ysmallincomparisonwiththe impulse

e locitygi entotheringismuchgreater



s s_ u s e # p d



M S

entra lpartso f thef ilm. Hence unless

theringissov erygreatastoreducethe

d forwardwithittosomethingnotincomparablygreaterthanthe v olumeofthesolidring itself the

lconf igurationsofmotionsw eha ebeen

bybutinsensibleq uantitiesthemomentum

Theva lueof thismomentumiseasily

onof Green sformula.Thusthe

ortionoffluid carriedforward( being

of thesamedensitymo ingw iththe

getherw ithane ui a lentforthe inertiao f the

s isappro imately thesameina llthese

oaGreen sintegra le pressingthew hole

Thee ualityo f theef fecti emomentumfordif ferentve locit iesof theringiseasilyv erif iedw ithout

otsogreatastocausesensiblede iations

portionoffluidcarriedforward.Thus

of thea iso f theportionof thef luid

nedbyfindingthe pointinthea isof

ocity ise ua ltotheve locityof thering.

heplaneof theringthatv e locityv aries

of aninfinitesimalmagnetonapoint

lyasthecubeof thedistancef romthecentre .

usoftheappro imatelyglobular

simplein erseproportiontothe

andtherefore itsmomentumisconstantfor

ering. Tothismustbeadded asw as

uantitye ua ltoha lf itsow namount asan

rtiaof thee ternalf luid andthesumis

mentumofthemotion.Henceweseenot

emomentumisindependentofthe

butthatitsamountisthesameasthe

orrespondingringelectromagnet.The

btainedbytheGreen sintegralreferred

uste plainedisnotconfinedtothe

ngspecia lly re ferredto butise ua lly

ngs ofanyform detachedfromone

roughoneanotherinanyw ay ortoasingle

reeandq ualityo f " mult iplecontinuity "



s s_ u s e # p d



ysoastoha enoend. Ine erypossible

hef luidate erypo int w hetherof the

luidf i ll inga llspaceroundit isperfectly

' sformulhew hentheshapeof thecore

ticin estigationnow e pla inedpro es

ntumofthewholefluidmotionagreesin

iththemagneticmomentofthe correspondingelectromagnet.Hence stillconsideringforsimplicity

o fcore le tthislinebepro ectedon

htanglestooneanother.Theareas

bta ined(toberec onedaccordingto

autotomic astheywillgenera llybe are

ntumperpendiculartothesethreeplanes.

sultwill beagoode erciseon" multiple

uthorisnotyetsuf f icientlyac ua intedw ith

leresearchesonthisbranchofanaly tical

therornota llthek indsof " multiple

tedareincludedinhis classificationand

in estigationinwhichathin solid

emo inginanydirectionthrougha

e motionpre iouslye citedinit re uires

te erypo inttobe inf inite ly smallin

sofcur atureof itsa isandw iththe

ny otherpartoftheci rcuitfromthat

entheeffecti emomeno f thew hole

ndforav orte w ithinfinitely thincore

mberofsuchvortices how e ernearone

edsimultaneously andthew holeef fecti e

anddirectionwillbethe resultantof

ntcomponentv orticeseachestimated

etheremar ableproposit ionthatthe

anypossiblemotioninan infiniteincompressiblefluidagreesindirectionandmagnitudewiththe magnetic

ingelectromagnetinHelmholt ' stheory.

ethemathematicalformulaee pressing

mentinthemoredeta iledpaper w hichhe

eto laybeforetheR oyalSociety .

rstoanyone eitherobser ingthe

ingsorin estigatingthetheory -What



s s_ u s e # p d



M S

si eof theringinanycase Helmholt ' sin estigationpro esthattheangularv orte v e locityof the

tslength orin erse lyasitssectiona larea .

electriccurrentinthe electromagnet

tely thinvorte core remainsconstant

thmaybealteredinthecourseof the

periencesbythemotionofthe fluid.

the largerthediameterof theringfor

trengthofvorte motionsinanordinary

reateristhew holek ineticenergyof the

sthemomentum andw ethereforesee

lmholt ringaredeterminatewhen

hof thev orte motionaregi en and

ineticenergyorthemomentumof thew hole

ceif afteranynumberofcollisions

lt ringescapestoagreatdistancef rom

ornearly f ree f romv ibrations itsdiameter

edordim inishedaccordingasithasta en

nenergy to theothers. A fulltheoryof the

omsbyele ationof temperature istobe

rinciple .

fe hibitingsmo e-ringsisasfo llow s: A largerectangularbo openatoneside hasacircularho leof6

ntheoppositeside.A commonrough

tcube orthereabout w il lansw erthepurpose

eof thebo isclosedbyastouttowelor

eet ofindia-rubberstretchedacrossit.

sidecausesacircularvorte ringtoshoot

therside . Thevorte ringsthus

hebo isf i l ledw ithsmo e. Oneof the

ofdoingthisistousetw oretortsw ith

o lesmadeforthepurpose inoneof the

llquantityofmuriaticacidisputinto

ndofstrongli uidammonia intotheother.

romtime totimetooneor otherof

cloudofsal-ammoniacisreadilymaintained

A curiousandinterestinge periment

esthusarranged andplacedeither

notherorfacingone anothersoasto

meetingf rom oppositedirections-orin



s s_ u s e # p d



NA MICS [ 1

tions soastogi esmo e-ringsproceedingin

erat anyangle andpassingone

nces.Aninterestingv ariationofthe

debyusingcleara irw ithoutsmo einone

siblev orte ringspro ectedf romitrender

ysensiblewhentheycomenearanyof

dingf romtheotherbo .



s s_ u s e # p d



T IO N .

oya lSocietyofEdinburgh V o l. xx v . 1869

pril 1867.

daugmented28thA ugustto12thNo ember 1868.

ofthepresentpaperhasbeen

hypothesis thatspaceiscontinuously

siblefrictionlessli uidactedonbyno

lphenomenaofe eryk inddependso le lyon

uid. B utIta e inthef irstplace as

on a f initemassof incompressible frict ionless f luidcomplete lyenclosedinarigidf i edboundary .

elmaybeeithersimplyor multiply

uentlyconsidersolidssurrounded

w hicha lsomaybeeithersimplyormultiplycontinuous. Itw il lnotbenecessary toe cludethesupposit ionthat

theouterboundaryo ersomefinite

notsurroundedby the li uid buteach

rroundedby the li uidornot andw hether

mustbeconsideredasaparto f thew hole

egi enatrest andletnoforce

heconta iningv esse l orf romthesurfacesof

eractonanyparto f it . Lettherebe

erfectly incompressible andof thesame

te itherperfectly rigid ormoreorless

efinedasamass continuouslyoccupyingspace whose

ononeanothere erywheree actlyinthedirection

ceseparatingthem.

egralederhydrodynamischenGleichungen Iwelcheden

prechen C re lle ( 1858 translatedbyTa it inPhil. Mag.

dt eausderA nalysissitus & amp c. C re lle (1857 . Seea lso



s s_ u s e # p d



ctorimperfecte lasticity. Someof thesemay

oserigidity andbecomeperfectly li u id

dmaybesupposedtoac uirerigidity

s.Let thesolidsacton oneanother

res f rictions ormutua ldistantactions

of " actionandreaction. Letmotions

andinthe li uid e itherby thenatura l

sor bythearbitraryapplicationof

elimited time.Itisof noconse uence

ha ereactionsonmatteroutsidethe

thattheymightbeca lled" natura lforces in

e ( whichadmitsactionandreactionat a

edarbitrarilyby supernaturalactionwithout

mlocution and atthesametime toconformtoacommonusage w esha llca llthemimpressedforces.

usnessastodensityofthe contentsof

sse l itf o llow sthatthecentreof inertiaof

idandsolidsimmersedini tremainsat

the integra lmomentumof themotionis

nandTait sNaturalPhilosophy ~ 297 the

fthecomponentsofpressureonthe

ra lle ltoany f i edline ise ua ltothetimeintegra lo f thesumof thecomponentsof impressedforcespara lle l

ua litye ists o fcourse ateachinstant

mpressedforces andcontinuestoe ist

o f the irt imeintegra ls a f tertheyha e

bse uentmotionof theso lids andof the

othem w hate erpressuremaycometo

ssel w hetherf romthef luidorf romsome

ntactwithit thecomponentsofthis

y f i edline summedfore erye lementof

esse l mustvanishfore ery inter a lo f

ressedforcesact. If f o re ample one

conta iningv esse l therewillbeanimpulsi epressureof thef luido era lltheresto f thef i edcontaining

sumof itscomponentspara lleltoany line

othe correspondingcomponentof the

ntrarytodesignatemerelydirectionalopposition and

dw ordopposite tosignifycontraryandinoneline .



s s_ u s e # p d



T I O N

solidonthepartof thissurfacewhich

a n d co n si d er o b li u e i mp u ls e o f an i n ne r m o i n g

lidsphericalboundary . B ut a f terthe impressedforcesceasetoact andaslongasthecontainingvesse lis

so lids the integra lamountof thecomponentof f luidpressureonit paralle ltoany line v anishes.

dtostopthewholemotionof fluid

2 is done if theso lidsarebroughttorestby

esonly thetimeintegra lso f thesums

eforces paralle ltoanystatedlines may

e ualandcontrary tothetimeintegrals

sofcomponentsof the initia tingimpressedforces( ~ 3 ) . B utw esha llsee( ~~ 19 21 thatif the

f inite ly large anda llo f themo ingso lidsbe

ingthe wholemotion theremustbe

nq uestionbetweenthetimeintegra ls

arydirectionsof theinitiatingand

s buttheremustbe( ~ 21 complete ly

betweenthetwosystems.

on henceforthIshallusetheun ualifiedtermimpulsetosignifyasystemof impulsi e forces to

narigid body.Thusthemostgeneral

toanimpulsi e force andcouple inplane

ordingtoPo insot ortotw oimpulsi e

g accordingtohispredecessors.F urther

pulseofthemotionatanyinstant in

hesystemof impulsi e forcesonthemo eable

teitfromrest oranyothersystem

alenttothatone if theso lidswerea llrigid

honeanother as forinstance the

eforceandminimumcouple.Theline

eforcew illbeca lledtheresultanta is

emomentof theminimumcouple( w hose

is line willbecalledthe rotational

sdef inedthetermsIintendtouse I

errorsthatmightbefalleninto remar

wholemotionsofsolids andli uidis

e d e fi n ed a s t he i m pu l se b u t ( ~ 4 i s e u a l



s s_ u s e # p d



rce-resultanto f " the impulse andthe

ertedonthe li uidby theconta iningvesse l

emotion:and thatthemomentof

motionroundthecentreof inertiaofthe

snote ua ltotherotationa lmoment asI

se ua ltothemomentof thecoupleconstitutedby" the impulse andthe impulsi epressureof the

e li uid. Itmustbeborne inmindthat

owe erdistanta llroundf romthemo eable

esse lmaybe ite ercisesaf inite inf luence

mentofmomentumofthewholemotion

nte ly large andinfinitelydistanta ll

tdoessoby infinitelyslowmotionthrough

f luid ande ercisesnof inite inf luence

solidsorof theneighbouringfluid.

ood ifforaninstant wesupposethe

tobenotf i ed butq uite f reetomo eas

Themomentumofthe wholemotion

bute actlye ua ltotheforce- resultanto f

andthemomentofmomentumofthe

ntreofinertia willbepreciselye ual

ecouplefoundbytransposingthe constituentimpulsi eforcestothis pointafterthemannerofPoinsot.

the immersedsolids andof thefluidin

hweshallcallthe fieldofmotion will

edifference whetherthecontaining

rle f tf ree pro ideditbe infinitelydistant

therefore essentia lly indif ferent

edorletitbef ree. Theformersuppositionismorecon enientinsomerespects the latterinothers but

ntto lea eanyambiguity andIsha ll

e formerina llthatfo llow s.

impulseofthemotion andits

ceandcouple accordingtothepre ious

shedfromthemomentum andthemoment

w holecontentsof thev esse l le tthevesse l

epressureonthe li uidw illa lw aysbe

nt inalinethrough itscentre which

a landcontrary totheforce- resultanto f " the



s s_ u s e # p d



T I O N

erefore w ithitw illconstitute ingenera la

ofthiscoupleandthecouple-resultantof

ua ltothemomentofmomentumof the

ntreofthe sphere( whichisthecentre

esse lbe inf inite ly large andinf inite ly

emo eablesolids themomentof

motionisirre le ant andw hatis

sthe impulseanditsforceandcoupleresultants asdef inedabo e.

a ti n g ( ~ ~ 1 0 1 2 , a n d pr o i n g

5 , a fundamenta lproposit ioninf luidmotionw illbe

of the impulse w hetherof themo eable

consideredorofv ortices.

ntumofe erysphericalportionof

re lati e ly tothecentreof thesphere is

oatanyoneinstantfore eryspherica lportion

sf irsttoberemar ed thatthemoment

tof the li uidw hichatany instant

spherica lspacecane periencenochange

teofchangevanishesatthatinstant ,

eonit( ~ 1 , be ingperpendicularto its

reprecise ly tow ardsitscentre . Hence if the

thematterinthe fi edsphericalspace

y themomentofmomentumof thematter

g e actlythatofthematter which

e la t er ( ~ ~ 2 0 1 7 1 8 t h at t h is b a la n ci n g is

e itheramo ingso lid oro f someof the

ofw hichspherica lportionspossessmoment

efi edspherica lspace butit isperfect

10 asw illbepro edin~ 15.

ethefo llow ingpure lymathematical

narynotationu v , w forthecomponentsof

nt ( x , y z ) .

di t io n ( l a s t c l au s e o f ~ 1 0 r e u i re s t ha t

b e a c om p le t e di f fe r en t ia l , a t w ha t e e r i ns t an t

artofthe fluidtheconditionholds.

I b e li e e f i rs t p ro e d b y St o e s i n hi s p a pe r " O n t h e

Motion andtheE uil ibriumandMotionofElasticSo lids "

Transactions 14thApril 1845.



s s_ u s e # p d



v d y + w d b e a c om p le t e di f fe r en t ia l

onofx , y z , throughany finitespaceof

t thecondit ionof~ 10ho ldsthroughthat

e s p r oo f of L e mm a ( 1 : - i r st f o r

w hethersub ecttothecondit ionof~ 10or

nentmomentofmomentumroundOX of

hitscentre at0.Denotingbyfff

pacew eha e

d d yd . .. .. .. .. .. .. .. .. 1 .

( d w /d y o & a m p c . d en o te t h e v a l u es a t 0 o f th e

eha e byMaclaurin stheorem

d j '

ememberingthat( dw / d ) o & amp c. areconstantsforthespacethroughw hichthe integrationisperformed

. y d w

- dy y2 d d y d + - f f y d d yd

d 0

pleintegralsv anish becausee ery

ussphere isaprincipa la is andifA

ntumofthesphericalv olumeroundits

hesecond

inthee pressionforL w ef ind

.. . .. . . .. . 2 .

ccordingtotheconditionof~ 10 and

of theinfinitesimalspherenowconsidered

nt ofspacethroughwhichthiscondition

emustha e throughoutthatspace

. . 3 ) ;

1 .



s s_ u s e # p d



T I O N

2 , l et

d ( 4 ;

onentmomentofmomentumround

n y sp h er i ca l s pa c e wi t h 0 in c e nt r e. W e h a e [ ( 1

- v z ) . .. . .. . .. . .. . .. . . ( 5 ,

oughthisspace( notnowinfinitesimal .

d y - d. p. ( 6 ;

d - d = . . .. . .. . .. . .. . . 6 ;

onwithreferenceto- inthesystem

I r s u ch t h at

sin. . . .. . . .. . . .. . . .. . . 7 .

5 to thissystemofco-ordinates w eha e

. . .. . . .. . . .. 8 .

ce isspherical w iththeoriginofcoordinatesinitscentre w emaydi ide it into inf initesimalcircular

s ha ingeachfornormalsectionan

hd anddpforsides.Integrating

erings w eha e

ause( isa single- a luedfunctionof thecoordinates. HenceL= 0 w hichpro esLemma( 2 .

ynamicalproposition statedat

forthepromisedproof le tR denotethe

tyof thefluidacrossanyelement do o f

i tu a te d a t ( x , y z ) ; a n d le t u v , w b e t h e

esultantve locityatthispoint sothat

. . .. . . .. . . .. . . .. . . . 9 .

ingthehollowsphericalspaceacross

edtisR d. dt andthemomentof



s s_ u s e # p d



ngmassroundthecentrehas forcomponent

omponentofthemomentofmomentum

e sphericalsurfaceatanyinstant t

R d o . .. . .. . .. . .. . .. . .. 0 .

1 o f ~ 1 2 a n d th e n ot a ti o n of ~ 1 4 w e ha e

ariationperunitlengthperpendicular

thatisdifferentiationwithreferenceto r

beingdirectionalrelati elytothe

rd i na r y po l ar c o -o r di n at e s r 0 * , w e h a e

Od. . . . .. . . .. . . .. . ( 11 .

f continuity foranincompressible li uid

f o r e e r y po i nt w i th i n th e s ph e ri c al s p ac e a n d

dTait A pp. B ]

amp c. . .. . . .. . . .. . . .. . . . 12 ,

w h er e S o de n ot e s a co n st a nt a n d S1 S 2 & a m p c .

rdersindicated.

2 rS - 3 r 2 + & a m p c . .. . .. . .. . .. . 1 ) .

thesisofthe mostgeneralsurface

sect a l andtesseralharmonics[ Thomson

thatdSi/ drisasurfaceharmonicof thesame

w aysunderstandd2/ d 2+ d2/ dy2+ d2/ d 2.



s s_ u s e # p d



wasstatedabo e( ~ 5 . Topro ethis

Inow proceed.

surfacestobe describedrounda

simmersedina li uid. Thesurrounding

1 perpendicularly andthereforew henany

eneratedby impulsi e forcesappliedtotheso lids

meterofthemomentumofthe matter

ceatthef irst instant mustbee actly

those impulsi e forcesroundthisline .

sline ofthemomentumofthematter

woconcentric sphericalsurfacesisz ero

nyso lid andpro idedthat if thereare

no impulseactsonthem.

w hatw eha edef inedas" the impulse

6 , w eseethatitsmomentroundany line is

momentumroundthesameline o fa llthe

alsurfaceha ingitscentrein thisline

rto whichanyconstituentofthe

still hold thoughthereareother

rhood andimpulsesareappliedto

mentsofmomentumofthoseonlywhich

ntoaccount andpro idednoneof themis

11 regardingf luidoccupy ingatany

lsurface areapplicablew ithoutchangeto

ingthe spaceboundedbyS becauseof

hatnosolidiscutbyS. Hencee ery

15 asfarase uation( 11 , maybenow

hnS butinsteadof ( 12 w enow ha e

7 6 , i f we d en ot e by T 1 T 2 & a m p c . a no th er

monics

+ & a mp c .

a mp c . ' . . . .. . .

reatestand smallestsphericalsurface

ha ingnoso lidsinit becausethroughall

becausethisw ouldgi e inthe integra lo f f low across

e afiniteamountofflowout oforintothe space

rationordestructionofmatter.



s s_ u s e # p d



T I O N

ndthee uationofcontinuitypro ethat

te ad o f ( 1 ) , w e no wh a e

* * * * * ( 1 5 .

mp c .

d s in .. . ( 1 6 .

d r

5 ne itheranymo eableso lids norany

stwithinanyfinitedistanceof Sall

amp c. musteachbeinf inite ly small: andtherefore

0. Thispro esthepropositionassertedin~ 5:

escannotha ez eromomentrounde ery

iteportionofspace w ithoutha ing

-resultanteache ualtoz ero.

o lidshasnotbeenta eninto

hemmaybeli uef ied( ~ 3) w ithoutv io lating

19. Tosa ecircumlocutions Inow def ine

of f luidha inganymotionthatitcouldnot

transmittedthroughitselffrom its

e ly forbre ity Isha llusethee pression

so lidoravorte , oragroupofsolidsor

o edmaybenowstatedin terms

w hichwerenotusedin~ 5 andsobecomessimply this: -The impulseof themotionofaso lidorgroup

dthesurroundingli uidremainsconstantas

sufferedfromtheinfluenceofothersolids

conta iningv esse l.

~ 6 , thatthemagnitudesofthe

ationalmomentoftheimpulseremain

tionof itsa isin ariable .

mof thestaticsofarigidbodyw emay

ceand couplealongandroundthe

a lresultantforcea longtheparalle ll ine

agreatercouplethe resultantofthe

couple andacouple intheplaneof thetw o

momente ua ltotheproductof the irdistance



s s_ u s e # p d



we maypassfromtheforce-resultant

heimpulsealongand rounditsa is

antandgreatermomentof impulse by

a ny p oi n t Q , n o t in t h e a i s ( ~ 6 o f t he

entis( ~ 18 e ua ltothemomentof

ntQ , ofthemotionwithinanyspherical

ascentre w hichenclosesa llthev ortices

lidsorv orticesw hicha lwaysk eep

of f initeradius orasinglebody mo ing

anha enopermanenta eragemotionof

obli uetothedirectionof theforceresultantoftheimpulse ifthereisa finiteforce-resultant.F or

sphericalsurfaceenclosingthemo ing

ha emomentofmomentumroundthe

y.

motionof translationw hentheforceresultanto f the impulsev anishes andtherew illbe fore ample

hapedli e thescrew-prope llero fasteamer

omogeneousli uid andsetinmotionby

ndiculartothea iso f thescrew .

sultantofthe impulsedoesnot

enomotionof translation ortheremaye en

tionoppositeto it. Thus fore ample

motion established( ~ 6 ) throughit w il l

a trest. A ndifatany timeurgedbyan

n thelineofthe force-resultantofthe

n itwillcommenceandcontinue

agemotionof translationinthatdirection a

m andthesameasif therewereno

ringis symmetrical.Ifthetranslatory

cyclicimpulse butlessinmagnitude

aryto thewholeforce-resultant

e ualandoppositetothe cyclic

nslationwithz eroforce-resultantimpulse-anothere ampleofwhatisassertedin ~ 24.Inthiscase

mmetrical orofanyothershape such

w hi c h t o f i i d ea s w e h a e s u p p os e d gi e n



s s_ u s e # p d



T I O N

antpressurein allsensiblyq uiescent

disappearsfrompineach ofthe

0 b e ca u se a s o li d i s e u i li b ra t ed b y e u a l

hetimeintegra l( 2 , w eha e

. 6 ;

( LMN denotethechangesintheforceandcouple-componentsof the impulseproducedby theco ll ision

f 2 dt , Y = & a mp c . Z = & a m p c .

( ) + f 2 dt , M l= & a m p c . N = & a mp c . .. .. .. .. 7 .

uiescentintheneighbourhoodofthe

o ingbodyorgroupofbodiesisinfinitely

sthatbeforethecommencementand

onw eha ef= 0 ate erypo into f the

dy. Hence fore eryva lueof trepresenting

nof theco ll ision theprecedinge pressionsbecome

dt Y = & a m p c . Z = & a m p c .

f d t M = & a mp c . N = & a mp c .' " ) ;

ntegra lchangeof impulsee periencedbya

inpassingbesideaf i edbodywithout

egardedasasystemof impulsi eattractions

erywhere inthedirectionof thenormal and

runito farea. B utitmustnotbeforgottenthattheterm binthee pression[ ~ 3 1( 5 ] forpproduces

, aninf luenceduringthecoll ision the integra l

pearsf romthee pression[ ~ 3 2( 7 ] for

sioniscompleted thatis( ~ 29 a f terthe

sedawaysofarasto lea enosensible f luid

odofthefi edbody.

m ~ 2 , w e s ee t h at w h en t h er e i s no

ga instthef i edbody andw henthe

fsolidspassesaltogetherononeside of

irectionof thetranslationw illbedeflected as

hole anattractiontow ardsthef i edbody



s s_ u s e # p d



ccordingas( ~ 25 thetranslationisin

seoroppositeto it. F or ineach

redby theintroductionofanimpulse

uponthemo ingbodyorbodiesasthey

) thetranslationbeforeandaf tertheco llisionis

impulse andisalteredin direction

silyunderstoodfromthe diagrams

presentsthef i edbody thedottedline

dsII , thedirectionsof theforce- resultant

i etimes andthefulla rrow-headsTT ,

tion.

A I S

2.

o theclassillustratedbyfig. 1.

withcyclicmotion( ~ 25 established

stotheclassil lustratedby f ig. 2 if the

ghthefluidin thedirectionperpendicular

ntrarytothecyclic motionthroughits

w esubstitutev orticesforthemo ingso lids

thatthetranslationisprobablyalw aysinthe

. Hence asil lustratedby f ig. 1 there is

sifbyattraction w henagroupofv ortices

i edbody. Thisiseasilyobser ed fora

bysendingsmo e-ringsona largesca le

t splan insuchdirectionsastopass

f i edsurface. A nordinary12- inchglobe

dhungbyathincord answersveryw ell



s s_ u s e # p d



T I O N

o f ~ 3 0 3 1 3 2 i s c l e a rl y a pp l ic a bl e t o

g b o dy o r t o a gr o up o f v o r t i ce s o r mo i n g

lwaysnearoneanother( ~ 2 ) , passingneara

i edboundary andbeing beforeandaf ter

a tav erygreatdistancef rome erypartof the

Helmholt ring pro ectedsoastopass

eof twow alls showsadef lectionof its

attractiontowardsthecorner.

force- resultanto f the impulse is asw e

37 , determinatew henthef low of the li uid

anysurfacecompletelyenclosingthesolids

b u t no t s o f r om s u ch d a ta e i th e r th e a i s

nalmoment asw eseeatoncebyconsidering

whichmayafterwardsbesupposed

tonbyaforce inany linenotthroughthe

inaplaneperpendicularto it. F orthisline

andthe impulsi ecouplew illbetherotationa l

tonof thesolidandli uid. B utthe

w i ll m o e e a c tl y a s it w o ul d i f th e i mp u ls e

eforceofe ua lamountinaparalle ll ine

sphere withthereforethissecondline

ro forrotationa lmoment. F oril lustrationof

nnglatentina li uid( w ithorw ithout

festbyactions tendingtoa lteritsa is or

uga lforcedueto it see~ 66andothers

pulse inanydirectionise ua ltothe

tmomentumofthemassenclosedwithin

ngalltheplacesof applicationofthe

iththato f the impulsi epressureoutwardson

matterenclosedbyS( w hethera ll l i u id

ly so lid iso funiformdensity its

a lto itsmassmultipliedintotheve locity

ofthespacewithin thesurfaceSsupposed

ea lw aysthesamematter andw illtherefore

malmotionofS thatistosay onthe

elocityinthedirectionof thenormalat

he impulsi e f luidpressure corresponding

ctualmotionfromrest beingthetime



s s_ u s e # p d



R T EX M O T IO N I l

dff

d y d

.

theothertw otermsof ( 3 oncesimply

ThomsonandTait A pp. A ( a ] toasurface

( N - c os a d o. .. .. .. .. 4 ;

sition anda lso o fcourse thatif therebe

va lueof thesecondmemberisz ero .

ingthemagneticandhydro inetic

esoleconditionthatat e erypointof

facethemagneticpotentialis e ualto

weconcludethat47rtimesthe magnetic

smwithinanysurface inthemagnetic

otheforce- resultanto f the impulseof theso lids

orrespondingsurface inthehydro inetic

irectionsof themagnetica isandof the

lsearethesame.F orthetheoryof

stingtoremar thatindeterminatedistributionsofmagnetismwithinthesolids orportionsoffluidtowhich

) w ereapplied ordeterminatedistributions

he irsurfaces maybefound w hich

ternaltothemshallproducethe same

-potential andthereforethesamedistributionofforceas thedistributionofv elocitythroughthewhole

henthemagneticforce intheinterior

hemannere pla inedin~ 48( 2 o fmy

agnetism itise pressiblethrougha ll

oef ficientsofapotentia l and onthe

eticsystemud + v dy+ w d isnotacomplete

ughthespacesoccupiedbythesolids

esultantforceandresultantflowholds

teriortothemagnetsandsolids inthe

ystemsrespecti e ly . B utif theother

ewithina magnet[ Math.Theoryof

1 orThomson sElectrica lPapers Macmillan 1869.



s s_ u s e # p d



f oot-note and~ 78 , publishedinpreparation

n Electro -magnets. ( st i l l inmyhandsin

completed , andw hicha lonecanbeadopted

-magneticmattertra ersedbyelectric

forcehasnota potentialwithinsuch

ee( ~ 68 thatdeterminatedistributionsof

oughspacescorrespondingto thesolids

emcanbefoundwhichshallgi e fore ery

rtra ersedbyelectriccurrentsornot a

agreeinginmagnitudeanddirection

etherofso lidorf luid atthecorresponding

csystem.Thisthoroughagreementfor

o-magneticanaloguepreferabletothe

ingbegunw iththemagneticana logoussystem

enienceforthedemonstrationof~ 3 8 w e

ethepurelyeloctro-magneticanalogue.

a us e d in a n ti c ip a ti o n i n ~ 3 7 ( 1 w e

4 2 4 ) f i nd t h e mo me n tu m of t h e wh o le m at t er

ore enso lidalone-atany instantw ithin

ermsof thenormalcomponentve locityof

f thissurface or w hichisthesame the

surface itse lf ifw esuppose ittovarysoas

ematter.

meof thespaceboundedbyany

.As yetweneednotsupposeV constant.

o -o r di n at e s of t h e ce n tr e o f gr a i t y. W e h a e

. . .. . .. . .. . .. . .. . .. . .. . .( 5 ,

hatthee pressionw ithinit istobeta en

S. Now asSvariesw iththetime the

ista enw ill ingenera lvary butthe

swhichite periencesatdifferentparts

ea inthe inf inite ly smallt imedt contributeno incrementsordecrementsto f f x 2dyd ] , asw eseemost

tobeasurfacee erywherecon e

Z 2 d t d ] . .. .. .( 6

i i d t6



s s_ u s e # p d



O R T EX M O T IO N 3 3

locityw ithw hichthesurfacemo esinthe

r ma l a t ( x , y z ) , w e h a e i n t he p r ec e di n g e p r es s io n

eoutwardnormaltoO X . Hence

] .

im itsindicatedby [ ] areclearly

otinganelementof thesurface suchthat

erthew holesurface. Thusw eha e

. .. . . .. . . .. . 7 .

isconstant thisbecomes

8 .

. . .. . . .. . . .. . 8 .

ce S istheboundaryofaportionof the

form densityunity withwhose

thex -componentmomentumofthis

n d t h e r ef o re e u a ti o n ( 8 i s t he r e u i re d

n.

( 7 a n d ( 8 a r e pr o e d m or e s ho r tl y o f

ay ticalprocessgi enbyPo isson and

ch s u b e c ts t h us i n s ho r t. L e t u v , w

ocity o fanymatter compressibleor

p o in t ( x , y z ) w i th i n S a n d le t c d en o te

of d u /d + d / d y + d w /d , s o t ha t

.. . . .. . . .. ( 9 .

omponentmomentum of thew holematter

yat theinstantconsidered

x d yd - -| d , d dy d . .... . 1 0 .

~ 60.

Magnetism.



s s_ u s e # p d



d

d d yd - x - + dd d d yd ,

J v J j J \ dy d j v

yd = ( v d d + wd d y .

anda lteringthee pressiontoasurface

n an d T ai t A p p. A ( a , w e h a e

u dyd + v d d + w d d y - ff fc d d yd

d y d . . .. . .. . .. . .. . .. . .. . .. . . 1 1 ,

7 .

mpressible w eha ec= 0by theformula

on" ofcontinuity ( ThomsonandTait

on ( 8 .

pretationofthedifferentialcoefficientsdu/d , & amp c. andofthee uationofcontinuity when asat

o f f luidandso lids u v , w arediscontinuousfunctionsha ingabruptlyvary ingva lues presentsno

pulseappliedtothecollision

orticesmo ingthroughali uid theforceresultanto f the impulsecorresponds asw eha eseen precise ly

mofasolid intheordinarytheoryof

aybefeltin understandinghowthe

4 o f thew holemassiscomposed therebeing

tumofsolidsandfluidsin thedirectionof

esnear theplaceofits application

C onsider fore ample thesimplecase

truc byasingle impulse inthe lineof its

inthedirectionof the impulse beforeand

thecontrarydirection inthespaceround

esofdi idingthew holemo ingmass

llustrati eof thedistributionofmomentum

ow ingproposit ions( ~ 45 w ithreferenceto

d ( ~ ~ 4 6 4 7 4 8 .

ro f f initeperiphery notnecessarily circular complete ly surroundingthev ortices( ormo ing

rsurroundingnone andconsiderthe in



s s_ u s e # p d



T I O N

riouslymo ingmatteratany instant

ylinders.Thecomponentmomentum

hef irst ise ua ltothecomponentof the

edirection andthatofthe secondis

herical surfaces oneenclosing

ingso lids andtheothernone. The

hewholematterenclosedbythefirstis

ulse andise ua lto2of itsva lue.

mofthewholefluidenclosedbythe

a llmo edw iththesamevelocity and

satitscentre .

planesat afinitedistancefrom

eld ofmotion butneithercuttingany

omponentperpendiculartothemof the

occupyingatanyinstantthe space

rthisincludesnone some ora llo f the

l id s i s z e r o .

ositions:Considerineithercaseafinite lengthoftheprisme tendingtoav erygreatdistanceineach directionfromthefieldof

byplaneorcur edends. Then the

aysuppose(~ 61 startedf romrestby

thesolids[ or( ~ 66 ontheportionsof fluid

s ; the impulsi e f luidpressureonthe

eratenomomentumparalleltothe

momentuminthisdirection therewill

mpressedimpulsi e forcesontheso lids andthe

esontheends butincase2therew illbe

pressureontheends. Now the impulsi e

dsdiminish[ ~ 50( 15 ] accordingtothe

distancef romthef ie ldofmotion w henthe

direction andarethereforeinfinitely

nfinitelylong eachway.Whence the

i c e p a ns i on s ~ 1 9 ( 1 4 , ( 1 5 , i n t he

3 7 ( 1 , ( 2 ; a n d t h e fu n da m en t al t h eo r em

0



s s_ u s e # p d



T h o ms o n an d T ai t A p p. B ( 1 6 ] a n d

ase andT i= 0fortheother w epro ethe

45 immediately .

~ 45 thew ell- now ntheoryofelectric

tor maybecon eniently referredto .

sthenormalcomponentforceatany

duetoanydistribution pa o fmatterin

eoftheplane adistributionofmatter

gN1/27rforsurfacedensityateachpo int

spthroughallthespaceontheotherside

ereforethatthewholequantityofmatterin

se ua ltothew holequantityofmatter

odenotingintegrationo erthe inf initeplane

. . .. . ( 12 ,

matterinp bez ero . Hence ifNbethe

throughspace onbothsidesofthe

wholequantityofmatteroneachside

.. . .. . . ( 1 ) ;

parts foreachofwhichseparately

ranslatedintohydro inetics showsthatthe

ssany inf initeplane iszeroate ery

lidsorv ortices. Hence andf romthe

ch(~ 3 ) w eassume thecentreofgra ity

twoinfinitefi edparallelplaneshas

perpendiculartothematanytime

o ingso lidiscutbye ither: w hichisP rop. III.

nd D u b. M at h . J o u r n al 1 8 49 L i ou i l le s J o u rn a l 1 8 45

ofElectricalPapers(Macmillan 1869 .

tica llyw ithease bydirectintegrationsshow ing( w hether

neco-ordinates that

w e h a e

d yd _

z 2



s s_ u s e # p d



T I O N

tteracross anysurfacewhate er

w holevo lumeof thef inite f i edconta ining

partsisnecessarily zero becauseof the

andthereforethemomentumofallthe

aralle lplanes e tendingtothe inner

esse l andtheportionof thissurface

mhasalwaysz eroforitscomponent

anes w hetherornotmo ingso lidsor

erorboththeseplanes. B utit isremar ablethatw henanymo ingso lidorv orte iscutbyaplane the

ssthis plane( ifthecontainingv essel

desf romthef ie ldofmotion , con ergestoagenerally f initev a lue astheplane ise tendedtov ery

omthe fieldofmotion whicharestill

onwiththedistancesto thecontaining

sf romthatf initev a luetozerobyanother

thedistancestow hichtheplane ise tended

parablewith andultimatelybecome

esof thecur e inw hichitcutstheconta ining

wit isthattheconditionof neither

gso lidorv orte isnecessary toa llow ~ 45

eferencetotheconta iningv esse l and

ua lity toz eroassertedinthisproposit ion

appro imatedtow hentheplanesare

a llround w hich thoughinf inite ly shorto f

a iningvesse l areverygreatincomparison

stancesfromthemostdistant partsof

concernedin~ 45 I. III. maybe

ulartotheresultantimpulsedrawany

esof thef ie ldofmotion w itha llthe

ticesbetweenthem anddi ideaportionof

tofiniteprismaticportions bycylindrical

erpendiculartothem. Supposenow oneof

includeall themo ingsolidsand

ta lteringtheprismaticboundary le tthe

edinoppositedirectionsto distanceseach

a incomparisonw iththedistanceof themost

o lidsorv ortices. B y~ 45 I. themomen



s s_ u s e # p d



thisprismaticspace is( appro imately

tant I o f the impulse andthatof the

theothersis( appro imate ly z ero .

a p p ro i m at e ly z e r o v a l ue s m us t o n

bee ualto - I if theportionsof theplanes

prismaticspacesbee tendedto

omparisonwiththe distancebetweenthe

s w eha eonly toremar thatifb

entialatapoint distantDfromthe

dx f romaplanethroughthemiddleperpendiculartothe impulse w eha e( ~ 5 ) appro imately

omparisonwiththeradiusof thesmallest

mo ingso lidsorv ortices. Hence putting

planesunderconsideration denotingbyA the

ftheprismaticportions andcallingD

forthisarea w eha e( ~ 45 forthe

tionwithinthisprismatic space

mo ingso lidsorv ortices

2isan infinitelysmallfraction( asa/Dis

sf inite ifA / D2isf inite pro ideda/Dbe

sintegra lva lue( compare~ 48 footnote con ergesto- w hentheportionofarea includedinthe

tilla/Dis infinitelysmallforallpoints of

mathematicaltheoryofthe con ergenceofdefiniteintegrals andasillustratingthedistributionof

isinterestingtoremar that udenoting

ra l le l t o x , a t a ny p o in t ( x , y z ) , t h e in t eg r al

p re ss in g mo me nt um m ay a s is r ea di ly p ro e d h a e

ooaccordingtotheportionsofspace

n.

thedistributionofmomentum

lbesphericaloffinite radiusa.



s s_ u s e # p d



O R T EX M O T IO N 3 9

9

a m p c . 1 4 ,

a mp c ..

pro idedrislessthana andgreaterthan

concentricsphericalsurfaceenclosingall

ow by thecondit ionthattherebeno

nta iningsurface w emustha e

a . .. . .. . .. . .. . .. . .. . .. 1 5 ,

1 = T ~ . 1 6 ;

a2i+1. . . .. . . .. . . .. . . .. . . .. . . . 16 ;

+ 3 I + 2 + & a mp c .. .. .. .. .. 1 7

] i f t he w h ol e a mo u nt o f t he x - c om p on e nt o f i mp u ls i e

fluidwithin thesphericalsurfaceofradius

tbedenotedbyF , w eha e

. .. . . .. . . .. . . .. . 18 ,

o f theradiusthroughdo. Now

cof thef irstorder andthereforea llthe

pansion e ceptthef irst disappearinthe

uentlybecomes

T ic o s 0 r. .. . .. . .. . .. 1 9 .

- . + . . . . .. . . .. . . .. . . .. ( 20 ,

n d Ta i t A p p. B , ~ ~ i j ] t h e mo s t ge n er a l

eharmonicof thefirstorder. Weha e

efore( byspherica lharmonics orby the

mentsofinertiaofauniformspherical

4 7r A

o - 3 . . . . .. . .. 2 1 ;

4 A .. . .. . . 2 2 .

= l + 2 - . . .. . .. .. . .. . .. . . 2 2 .



s s_ u s e # p d



ethex -momentumof thef luidatany instant

oncentric sphericalsurfacesofradius

.. . .. . .. . ( 2 ) .

itely smallincomparisonw itha this

asitoughttodo inaccordancewith~ 45 II.

4. 47rA

inginthefluid outsidethesphericalsurface

ua landoppositetothat( ~ 45 II. o f

therf luidorso lid w ithinthatsurface.

n d ~ 5 2 w e se e t ha t i f X , Y Z b e

ftheforce-resultantoftheimpulse the

ce pansion( 14 isasfo llows: T - -2X x + Yy+ Z z . . . .. . . .. . . . (25 ,

andv orticesta enintoaccountarewithin

adiusis v erysmallincomparisonwith

orticesormo ingso lids andw iththe

edboundingsurface.

splendidpaperonV orte Motion has

ntremar thatacerta infundamenta l

w hichhasbeenusedtodemonstratethe

onsinhydro inetics issub ecttoe ceptionw henthefunctionsin o l edha emultiplev a lues. Thisca lls

nde tensionofe lementaryhydro inetic

proceed.

m( 1 o fThomsonandTait A pp. A

d+ d __

dr + d od o d d yd

dy d d C u

d O V 2 0 t= f d f - b ff -d d yd , ' V , 2 .. .. .. .. .( 1 ,



s s_ u s e # p d



T I O N

ceptionif4andb denoteany tw osingle a luedfunctionsofx , y z ; f fd dyd integrationthroughthe

niteclosedsurface S f do integration

rface andbrateofvaria tionperunito f

ctionatanypo into f it . ThisisGreen s

Helmholt ' sl im itationadded( inita lics .

forhimself.

isamany - a luedfunction andthe

c/ d , . .. d d / d , . .. e a ch s i ng l e- a l ue d

1 cannotbegenera lly true. Itsf irst

mbiguous buttheprocessofintegration

beror thethirdmemberisfound would

rO ismany- a lued. Inonecasethe

ote ua ltotheambiguoussecond w ould

p r o i d ed b i s n ot a l so m a ny - a l ue d a n d in

mber thoughnote ua ltothethird w ould

pro idedf isnotmany - a lued.

' = tan- Y . . . . . .. . . .. . . .. . . .. . . .. . . .. 2 ,

tionsoftwo planesperpendicularto

tw eentw ocircularcy lindersha ingOZ for

ionsof thesecylindersinterceptedbetw eenthe

dricalboundarye cludesfromthe

elineOZ w hereO ' hasaninfinitenumber

/d , a nd d o / d h a e i nf in it e v a lu es . W e ha e

x Y . .. . .. . .. . .. . .. . .. . . 3 ) ,

2 y2 .( " 3 )

b / ' = 0 . T h en i f 4 b e si n gl e - a l ue d

rocesspro ingthee ua litybetw een

bersof ( 1 , w hichbecomes

d x d yd = o . .. .. .. .. .. .. .. 4 .

t o e nd .

becomes

2 d d y d . .. . .. . .. 5 ,

J x ' v v



s s_ u s e # p d



biguousintegrationofthefirstmember

edbyS asw eseebye amining inthis

aningofeachstep oftheordinaryprocess

esforpro ingGreen stheorem. It is

ddto( 5 aterm

alueof tan-ly / isrec onedcontinuously

eplaneZ OX totheother: or

O

onesideofZ O Ytotheother torender

st m e mb e r of ( 1 . T h us t a i n g fo r

rmof theaddedterm w enow ha eforthe

to n ( 1 f o r th e c as e o f b = t a n- l y / , b a n y

andSthesurface composedof thetwo

oparalle lplanesspecif iedabo e:

d

0 27 x d ( d ~ f d ot an -Y -Y ' 1 + y 2 i \ d y/ y= O X

. .. . . .. . . .. . . .. . . . 6 .

nybarrierstoppingcirculationround

allambiguitybecomesimpossible and

1 ho lds. F orinstance if thebarrierbe

O X , interceptedbetw eentheco-a a l

nesconstitutingtheSof~ 55 sothat

integrationo ereachsideof thisrectangular

ssimply thestrictapplicationof ( 1 tothecase

ceptiona linterpretationofGreen s

asese emplif iedin~~ 55and56 depends

ayha edif ferentva luesw henrec oned

entcur es draw nwithinthespace

po intP toapo intQ; dsbe inganinf initesimale lementof thecur e andF therateofv aria tionofGper

tPC Q, PC Q betw ocur esforw hich

a lues andletbothliew hollyw ithinS.



s s_ u s e # p d



T I O N

omPtoQ; ma eitf irstco incidewith

y itgradua llyuntil itco incidesw ithPC Q ; it

ediateformscuttheboundingsurface

dy + d

ta inedw ithinS anddc/ d , do / dy do/ d

uousbyhypothesis whichimpliesthat

esfora llgradua lvaria tionsofonecur e

chly ingw hollyw ithinS. Now inasimply

r e j o iningthepo intsPandQ maybe

nycur ePC Q toanyotherPC Q , and

ainedwithinS besimplycontinuous

nthemultiplicityofv a lueofGorO'

ermultiplycontinuous( . 58 thespace

maybee adedifw eanne toSasurface

eryapertureorpassageonthe opennessof

itydepends fortheseanne edsurfaces

nospace donotdisturbthetriple

dw ill therefore nota ltertheva luesof itsf irst

ingthemultiplicityo fcontinuity they

sby parts bywhichitssecondor third

f romallambiguity . Toa o idcircumlocution w esha llca ll/ theaddit ionthusmadetoS andfurther

s( ~ 58 notmere lydoublybuttriply

remultiply continuous w esha lldesignateby

/ ; a nd s oo n t he s e e r a l pa rt s of / r e u ir ed

iplecontinuityofthe space.These

tedetachedf romoneanother asw henthe

ueto detachedrings orseparate

utonepart/ maycutthroughpart

he n t wo r i ng s ( ~ 5 8 d i ag r am l i n e d i nt o o ne

constitutepartoftheboundaryof the

sha lldenoteby J ' ds integrationo er

r a ny o n e of i t s pa r ts / , / 2 & a m p c . L et n o w

lynearapo intB , o f / butonthetwo

c denotetheva lueof f dsa longany

espaceboundedbyS andj o iningPQ

r thisva luebeingthesamefora ll

ra llposit ionsofB tow hichitmaybebrought



s s_ u s e # p d



andw ithoutma inge itherPorQ passthrough

say K cisasingleconstantw henthe

ublycontinuous butitdenotesoneor

C 2 . . . Kmc w hichmaybealldifferentf rom

space isn-plycontinuous. Lastly le tK'

t r e la t i e l y to 4 a s K c r e l a ti e l y to b .

psoftheintegrationsbyparts now

biguity theaddit ions

f f dS b f. . .. . .. . .. . .. . . 7 ,

embersof ( 1 : 2denotingsummation

edif ferentconstituents1 2 / 2 . . . o f /

enthespace is( ~ 58 notmorethan

stheoremthuscorrectedbecomes

bd b ' \ d y d

fd sb ff - f 0 2 d d yd

f - j l V 2 d d yd . .. .. . 8 .. .. .. ) .

ogyofR iemann ask now ntome

Isha llca lla f initeposit ionofspacen-ply

ingsurfaceissuch thattherearen

enanytwopointsinit. Topre ent

Iadd( 1 , thatbyaportionofspaceImean

o into f itmaybetra e lledto f romany

tcuttingtheboundingsurface ( 2 , that

nofa ll l iew ithintheportionofspacereferred

hatby irreconcilablepathsbetw eentw opo intsP

such thata linedraw nf irsta longoneof

changedtillit coincideswiththeother

ingthroughPandQ andalwaysw holly

considered.Thus whenallthepaths

ereconcilable thespaceissimply

are j usttw osetsofpaths sothateach

withanyoneofthe otherset the

s whentherearethreesuchsets itis

soon. Toa o idcircumlocutions w esha ll

daryofahollowspacein theinterior

thatnooperationsw hichw esha llconsider



s s_ u s e # p d



T I O N

peningtothespaceoutside it. A tunnel

at eachendintotheinterior space

edoublycontinuous andifmore

erynew oneaddsonetothedegreeof

onesuchtunnelhasbeenmade the

ntinuouswiththewholebounding

sidered andinrec oningdegreesof

nse uencewhethertheendsofany f resh

otherofthis wholesurface.Thus if

y side aholeanywhereopeningfrom

raddsonetothe degreeofmultiple

hedfromtheouterboundingsolid

dormo able inthe interiorspace addsto

olatedportion butdoesnotinterfere

ult iplecontinuity. Thus ifw ebegin

paceboundedoutsideby theinner

ternalso lid andinterna llyby the

solidin itsinterior andifwedrill a

cedoublecontinuity.Twoholes or

achwithonehole( suchastwoordinary

tutetriplecontinuity andsoon. A spongeli eso lidw hoseporescommunicatew ithoneanother i l lustratesa

ontinuity andit iso fnoconse uence

thee ternalboundingso lidorisan

. Anothertypeofmultiplecontinuity

gslin edinoneanother w asreferred

edintooneanotherinv arious

ecomplicatedmutualintersectionsofthe

/ , 2 . . . re uiredtostopa llmult iple

a inganyportionof thebounding

nthatcase inw hichoneatleasto f thetw o

ev arietiesofmultiplecontinuitycuriously

edbya singleordinarystraightor

sufficientlybythesimplesttypes which

unnelalongalineagreeingin formwith

reonw hichasimplek notist ied andby

w irew ithak notonitto thebounding

ceof thew ireshallbecomeparto f the

paceconsidered thek notnotbe ing



s s_ u s e # p d



ebeingarrangednot totouchitselfin

gak nottedw ire w ithitsendsunited in

Noamountofk nottingork nitt ing

nthecordw hosea isindicatesthe lineof

nywaythecontinuityofthe space

esimplicityof thebarriersurfacere uired

utit isotherwisew henak nottedor

fthe boundingsolid.Asinglesimple

gonlydoublecontinuity re uiresacuriously

ppingbarrier:which initsformof

utifully show nby the li uidf i lmadhering

ethef irstf igure dippedinasoapso lution

omplicationof thesetypes oro fcombinationsof themw ithoneanother e ludesthestatementsand

N o . - D e c. 1 8 69 [ ~ 5 9 -~ 6 4 ( / ] .

namica llemma forthe immediate

applyGreen scorrectedtheorem( ~ 57 to

roughamultiplycontinuousspace.

dby ittov erysimpledemonstrationsof

entaltheoremsofvorte motion andshall

asubstituteforthecommone uations

sf initetube o f inf initesimalnormal

fullo f l i u id( w hethercirculatinground

withitsendsdone awaybyunitingthemtogether.



s s_ u s e # p d



T I O N

isa lteredinshape length andnormal

andw ithanyspeed. Thea erageva lueof

o f thef luida longthetube rec oneda ll

pecti e lyof thenormalsection , v aries

of thecircuit.

s considerf irstasingleparticleofunit

force andmo inga longasmoothguiding

edandbentaboutquitearbitrarily . Letp

ure andA rthecomponentve locit iesof

w ardsthecentreofcur ature andperpendiculartotheplaneofcur ature atthepo intP throughw hich

assingatany instant. Let4bethe

eparticleitself alongtheinstantaneous

hroughP . Thus: V , 4arethree

fthev elocityoftheparticleitself.Let

hedirectionof4 o f thew holeforceon

m en t ar y k i n e t ic s

+ 7. .. . . .. . . .. . . ..

h ithertopublished ) w il lbegi eninthesecondv o lumeof

uralPhilosophy.Itmaybe pro edanalyticallyfromthe

emotionofaparticlea longav ary ingguide-cur e( Walton

ournal 1842 F ebruary ; ormoresynthetica lly thusLet1 m nbethedirectioncosinesofPT thetangenttotheguideatthepo int

e ispassingatany instant ( x , y z ) theco-ordinatesof

y z ) i t s co m po n en t v e l o c it i es p a ra l le l t o fi e d r ec t an g ul a r a e s .

a nd Z = 1 + ni m+ n ,

y+ n + i + 1 i = + i = + i +

d( ThomsonandTait sNaturalPhilosophy ~ 9 tobemade

intinasecondedit ion thattheangularve locityw ith

n i s e u a l to / i 2 + i Z 2 + - i 2 a n d i f t hi s b e de n ot e d

f the lineP perpendiculartoPT intheplane inw hich

ndontheside towardswhichitturns.Hence

ntv e locityofPa longP . Now if thecur ew eref i edw e

by thek inematicdef init ionofcur ature(ThomsonandTa it

e inw hichPTchangesdirectionwouldbetheplaneofcur ature.

supposed thereisalsoin thisplaneanadditionalangular



s s_ u s e # p d



YNA MIC S [ 2

usofcur ature andd: /ds dr/ dsrates

f rompointtopointa longthecur eat

adofasingleparticleofunitmass le tan

, o fa li uid f i l l ingthesupposedendless

twbe theareaofthenormalsection of

re / a is andSsthe lengtha longthetube

t atany instant sothat( asthedensity

,

denotetherateofvaria tionof thef luidpressure

t

1

. .. . ( 2 .

thetwoendsof thearcasmo ew iththe

t he k i n e m at i cs o f a v a r y in g c ur e

.. . . .. . . .. ( 3 ) ;

= s s d d . . .. . .. . .. . .. . .. ( 4 .

/ d t it s v a l u e b y ( 2 w e ha e

p d \ )

s

= ( I q 2 _ p . . .. . .. . .. . .. . .. . .. . . 5 ,

8 l 2 . ( 5 ,

s andacomponentangularve locity intheplaneofPTandV ,

othenormalmotionof thev ary ingcur e . Hencethew hole

resultantoftwocomponents

planeofr7.

) d= K

f t he t e t i s p ro e d .

f t he t e t i s p ro e d .



s s_ u s e # p d



T I O N

tf luidve locity and8thedifferencesfor

s.Integratingthisthroughthelength

hefluid itsendsP1 P2mo ingw ith

-p 2 - Q -p \ ... ... .. ( 6 ,

heva luesof thebrac etedfunction atthe

ecti e ly andE2denotingintegrationa long

now P2bemo edforward orP ibac w ard t i l lthesepo intsco incide andthearcP1P2becomesthe

tE denoteintegrationroundthewhole

comes

sremainsconstant how e erthetube

ropositiontobepro ed asthe" a erage

is f o un d b y di i d in g ( 8 s b y t he l e ng t h of

maginedinthepreceding hashadnoother

by itsinnersurface normalpressureonthe

cethe proposition atthebeginning

f romw hich asw eha eseen thatproposit ionfo llow simmediate ly

aterease andnotmere ly foranincompressible f luid butfor

sityisafunctionof thepressure bythemethodof

ordinatesfromtheordinaryhydro inetice uations.

D iD d

d d '

ria tionperunito f t ime o fany functiondependingon

gw iththef luid andw = J dp/ p pdenotingdensity . In

natesweha e

w .

x

t + & a m p c .

m p 4 - 5 , an d D= 0 w.

uationsreducetheprecedingto

d dy d - . .

t ion e uation( 6 genera lisedtoapply tocompressible f luids.



s s_ u s e # p d



y closedringoffluid formingpartofan

e tendinginalldirections throughany

andmo inginanypossibleway andthe

6 areapplicabletoany infinitesimalorinf inite

tmet. Thusinw ordsPR OP. ( 1 . The line- integra lo f thetangentia lcomponentv e locity

ofamo ingf luidremainsconstantthrough

T h e ra t e of a u gm e nt a ti o n p e r un i t of t i me o f

elocityalonganyterminatedarcofthe

e s s o f th e v a l u e of q 2 - p a t t he e n d

velocity isrec onedasposit i e abo e

nd.

t ha t u d + v d y + w d i s a c om p le t e

bo e( ~ 1 ) tobethecriterionof irrotationa l

60 ( a ] isthesamein alldifferent

sfromonetoanotherof anytwopoints

histhesamething

60 ( a ] isz erorounde eryclosedcur e

dtoa pointwithoutpassingoutofa

which thecriterionholds.

j u s t pr o e d w e se e t ha t t hi s c on d it i on

ranyportionofamo ingf luidforw hich

andthusw eha eanotherproofof

t h eo r em ( 1 6 g i i n g us a n e w v i e w of i t s

w hich[ seefore ample~ 60( g ] w esha ll

nthetheoryofv orte motion.

naclosedcur e capableofbe ingcontractedtoapo intw ithoutpassingoutofspaceoccupiedby

uid thatthecirculationisnecessarily

otion. In~ 57w esaw thatacontinuous

doublyormultiplycontinuousspace may

tona lly yetsoastoha ef initecirculation

Q ' P p r o i d ed P P Q a n d PQ ' Q a r e " i r r ec o nc i l ab l e pa t hs b e tw e en P a nd Q . T h a t t he c i rc u la t io n mu s t be t h e

cilableclosedcur es( compare~ 57 , is

ncefromthenow pro ed[ ~ 59( P rop. 2 ]



s s_ u s e # p d



T I O N

60( a ] ina llmutua lly reconcilableconterminousarcs. F orby lea ingoneparto faclosedcur eunchanged andvary ingtheremainingarccontinuously nochange

inthispart and by repetit ionsof the

emaybechangedtoanyotherreconcilable

ntarypropositions.( a Thelineintegralofthetangentialcomponentv elocityalonganyfinite

d inamo ingf luid isca lledthef low inthat

thatis if it f o rmsaclosedcur eor

ow isca lledcirculation. Theuseof theseterms

mentsofP roposit ions(2 and( 1 o f~ 59to

Prop. ( 2 ] . Therateofaugmentation perunito f t ime

tedlinew hichmo esw iththef luid is

f thev a lueof~ q 2-pattheendf romw hich

eendtow ardsw hich posit i e f low isrec oned.

] . T h e ci r cu l at i on i n a ny c l os e d li n e mo i n g

constantthroughalltime.

esurface ly ingaltogetherw ithinaf luid

raw nacrossit thecirculationinthe

e ua ltothesumof thecirculationsin

rs. Thisisob ious asthe lattersum

sit i eandnegati e f low ineachportionof

woparts addedtothesumof theflows

hesingle boundaryofthewhole

ulationroundtheboundariesof inf initesimalareas inf inite lynearoneanotherinoneplane aresimply

s.

anyparto f thef luidrotateasaso lid

ngingshape ; orconsidersimply therotation

ion intheboundaryofanyplanef igure

a ltotw icetheareaenclosed multipliedby

e locity inthatplane( orroundana is

e . F or ta ingr 0todenotepo larcoordinatesofanypointintheboundary A theenclosedarea and



s s_ u s e # p d



elocity intheplane andcontinuing

w e ha e . r d

o O r 2 3 = o x 2 A .

o r a f l ui d m o i n g in a n y ma n ne r t h e

daryofan infinitesimalplanearea

area isca lledthecomponentrotationin

a isperpendiculartothatplane o f the

inglew ord" rotation isusedfor

ton: andthedef init ionisj ustifiedby ( c

1 ( 2 a bo e a pp li ed t o ( p b el ow .I t ag re es

w iththedef init ionof rotationinf luidmotion

le e b y S to e s a n d us e d by H e lm h ol t i n

Motion a lso inThomsonandTa it sNatural

a n d 19 0 ( j ) .

7 ' b e t he c o mp o ne n ts o f r ot a ti o n at

id roundthreea esatrightanglestoone

mponentroundana is ma ingw iththem

1 m n

neperpendicularto thelast-mentioned

einA B , C . Thecirculationinthe

B C i s b y ( b e u a l to t h e su m of t h e

riesPB C PC A andPA B . Hence

ry theareasof thesefourtriangles w eha e

.

e th e p ro e c ti o ns o f A o n th e p la n es o f t he p a ir s o f

andsotheproposit ionispro ed.

thatthecomposit ionof rotationsina

ompositionsofangularv elocitiesofa

ti e s o f f or c es & a m p c .

nf initesimalparto f thef luid the

peripheryofe eryplaneareapassing



s s_ u s e # p d



T I O N

posit ion andthereforeu v , w maybe

ofx , y z . Inapure lyana ly tica ll ight

ntbearingonthetheoryofthe integrationofcompleteorincompletedifferentials.Itwasfirst gi en

oreanalyticalproofthanthepreceding

N a tu r al P h il o so p hy ~ 1 9 0 ( j ) .

h ( j ) ( n ( o o f th e pr es en t se ct io n ( ~ 6 0

andw ithhisintegrationforassociated

ionalmotioninan unboundedfluid to

stitutehisgenera ltheoryofv orte motion.

p ur el y k i ne ma ti ca l ( h a nd ( j ) a re d yn am ic al .

ca llacircuitanyclosedcur enot

point inamultiplycontinuousspace.

s any twosuchclosedcur esif

58 butdif ferentmutually reconcilable

calleddifferentcircuits.

p l y co n ti n uo u s sp a ce i s a s pa c e fo r w hi c h

n dif ferentcircuits. Thisismerely the

bre iatedby thedef initeuseof theword

pose.Thegeneralterminologyregarding

nuousspacesis asIha efoundsince

ogetherduetoHelmholt ; R iemann ssuggestion tow hichherefers ha ingbeenconf inedtotw o-dimensional

edsomew hatf romtheformofdefinit ion

mh o lt , i n o l i n g a s i t do e s t h e di f fi c ul t

arrier andsubstitutedforitthe

ndirreconcilablepaths.Itis noteasy

gbarriero fanyoneof thef irstthree

tounderstanditssingleness butit iseasy

ethreecases any twoclosedcur es

erepresentedinthediagramsarereconcilable accordingtothedef init ionof thistermgi enin~ 58 and

conceptionw ecanma enouseof thetheoryofmultiple

tics( see~~ 61-6 ) , andHelmholt ' sdef init ionis therefore

ltothat whichIha esubstitutedforit.Mr Cler

. B . List inghasmorerecently treatedthesub ectofmultiple

pletemannerinanarticle entitled" DerCensusriumlicher

. Ges. Gottingen 1861. Seea lsoProf . C ay ley " O nthePartit ion

1861.



s s_ u s e # p d



enceofanysuch solidaddsonlyoneto the

space inwhichitis placed.

rt it ion asurfacew hichseparatesa

rs and ashitherto abarrier anysurface

f thespace Helmholt ' sdef init ionof

estatedshortly thus: A space is( n+ l plycontinuousifnbarrierscanbedraw n

hisapartit ion.

aspo intedoutthe importance inhydro- . ineticsofmany- a luedfunctions suchastan- ly / , w hichha e

fgra itation e lectricity ormagnetism

presse lectro-magneticpotentia ls andthe

hepartof thef luidwhichmo esirrotationally invorte motion. It is therefore con enient before

shouldf i uponatermino logy w ith

hatk ind w hichmaysa euscircumlocutionshereaf ter.

, y z ) w i ll b e c al l ed c y cl i c if i t e p e ri e nc e s

e ery timeapo intP o fw hichx , y z are

ordinates iscarriedfromanyposition

hesamepositionagain withoutpassing

hiche itherdb/d , d4/ dy ordo/ d

alueofthisaugmentationwillbecalled

particularcircuit.Thecyclicconstant

amevaluefora llcircuitsmutua lly reconcilable( 58 inspacethroughoutw hichthethreedif ferential

e.

ctioniscyclicw ithreferencetose era l

cilablecircuits itiscalledpolycyclic.

onesetofcircuits it isca lledmonocyclic.

tareaofacircleas seenfromapoint

y w he r e in s p ac e i s a m on o cy c li c f un c ti o n of x , y z , o f

47r.

e cur eofthe( 2 n thdegree

osed(thatisf initeendless branches

eenclosedw ithinothers isann-cyclic

n-cyclicconstantsareessentia llye ua l



s s_ u s e # p d



T I O N

m on g t hr e e v a r i ab l es ( x y z m a ye a si l y

uouscur es constitutingoneormore

essbranches( w hichmaybek notted asshown

of~ 58 orlin edintooneanother as

Theintegra le pressingw hat forbre ity

areaofsuchacur e isacyclicfunction

hasessentia llye ua lv a luesfora ll itscyclic

rentareaofaf initeendlesscur e( tortuous

thesumof theapparentareasofa llbarriers

andraw withoutma ingapartit ion.

erypolycyclicfunctionmaybe

ocyclicfunctions.

nisca lledcyclicunlessthecirculationis

paththroughthef luid w henit iscalled

s( e essentia lly cyclic.

t io n ma y [ ~ 5 9 ( f ] b e e it h er a c yc l ic o r

yclicifthereis onlyonedistinct

therearese era ldist inctcircuits inw hich

relycyclicif theboundaryofthe

nallymo ingfluidisat rest.Ifthe

hemotionof thef luidiscyclic it isacyclic

preparedto in estigatethemostgenera l

nofasinglecontinuousfluid mass

rmultiplycontinuousspace w ith for

dary anormalcomponentve locitygi en

nly totheconditionthatthew holevo lume

licmotion. Commencing asin~ 3 , w ith

hout letallmultiplicityofthe continuityofthespaceoccupiedbyit bedoneawaywithby

es 8i 3 2. . . stoppingthecircuits as

boundingsurfaceof thef luid w hich

ner surfaceofthecontainingv essel

tendedtoincludeeachside ofeachof

asin~ 3 , anypossiblemotionbe

boundingsurface. The li uidisconse uently setinmotion pure ly throughf luidpressure andthe

5 or60 59 throughoutirro tationa l. Hence



s s_ u s e # p d



gtheprescribedsurfaceconditionsis

lmotionis o fcourse( astheso lutionof

unambiguous. B utf romthisbarephysica l

ensuspect w hatthefo llowingsimple

uationpro es thatthesurfacenormal

terminestheinterior motionirrespecti ely

of themotionf romrest.

f irro tationa lmotioninsimplycontinuousspace. In~ 57( 1 , w hichisimmediate lyapplicable as

l y co n ti n uo u s m a e 0 = p a n d pu t

maybethev e locitypotentia lo fanincompressible f luid. Thatdoublee uationbecomesthefo llow ing

y d = b

ionffdomustnowincludeeach sideof

c s i , 2 . .. . H en c e i f b o = 0 f o r e e r y

face w emustha e

d yd = O ,

snomotionof theboundarysurface in

a therecanbenomotionof the

terior whenceitfollowsthatthere

ernalirrotationalmotionswiththe

ponentv elocities.Thus asaparticular

luidatrest le titsboundarybesetin

gaintorestatany instant a f terha ing

anye tent throughanyseriesof

dcomestorest atthatinstant.

portanttheorem whichdiffers

ding andincludeswhatthepreceding

lyanalyticalproofofthe possibilityof

houtthefluid fulfillingthearbitrary

dabo e aswasfirstpublishedin Thomson

so p hy ~ 3 1 7 ( 3 ) , a n d is t o b e gi e n



s s_ u s e # p d



T I O N

riationande tension. Inthemeantime

oursel esastothepossibilityo f irrotationa l

rioussurface-conditionswithwhichwe

ethesurfacemotionsarepossibleandre uire

d [ ~ 1 0 -1 5 o r ~ 5 9 b e ca u se t h e fl u id c a nn o t

nthroughfluidpressurefromthemotion

goon bya idofGreen se tendedformula

t o p ro e t h e de t er m in a te n es s o f th e i nt e ri o r mo t io n

specifiedfor multiplycontinuous

o ne b y h is u n al t er e d fo r mu l a [ ~ 5 7 ( 1 ] f o r

onalmotion.In thecaseof

61 theva lueof thenormalcomponent

entlyarbitraryo erthewholeboundary

a l u es p o si t i e a n d ne g at i e o n t he t w o

ers/ , 32 & amp c. Wemustnow introduce

rthat w henthebarriersare li uef ied

ybeirrotationalthroughoutthespace

lecontinuity.F oralthoughweha e

omponentv e locity ise ua le erywhere

arrier w eha ehitherto le f tthetangentia lv e locityunheeded. If theyarenote ua lonthetwo

direction therewillbeaf initeslippingof

rfaceleftbythedissolutionof the

mbrane constituting[ ~ 60(m abo e ,

a" v orte sheet. Theanaly tica l

t ionofe ua litybetw eenthetangentia l

aria tionof theve locitypotentialin

bee ualonthetwosides ofeach

ration weseethatthedifferencebetween

ocitypotentia lonthetw osidesmustbethe

eachbarrier. Thiscondit ionre uiresthat

e ua lo erthew holemembrane. F or

titutingof themotion le tpl P2bethe

P2of thef luid andmo ingw iththe

anotheronthe twosidesofoneof the

hepressurew w hichmustbeappliedtothe

differenceoffluidpressure onthetwo

-p2inthedirectionopposedtop . A ndlet



s s_ u s e # p d



itypotentialsatP iandP2 sothatif fdsdenote

a longanypathP IPP2w hate erf rom

oughthef luid( andthereforecuttingnone

and~ thecomponentof f luidve locitya long

f thiscur e w eha e

.. . . .. 1 .

59

. .. . .. . .. . .. . .. 2 ,

notetheresultantf luidv e locit iesatP IandPi.

nentv elocitiesatPiandP2 arenecessarily

if thecomponentspara lle lto thetangent

gmembranearea lsoe ua l w eha e

s

mponentve locit iesatP iandP2arenot

e sa me d i re c ti o n 0 2 - 0 m us t a s w e ha e

erthemembrane andthereforew musta lso

ssurehasbeenapplied foranytime

ofuniformvaluea llo erthemembrane

pliedno longer andthemembrane( ha ing

isdoneawaywith.Thefluid massis

tateofmotion w hichisirro tationa l

Th e " c i r cu l at i on [ ~ 6 0 ( a ] , o r t he c y cl i c

2-01 fore erycircuitreconcilablew ith

e uation

. . (4 ,

ale tendedthroughthewholeperiod

te v alue.

onmaybeperformed oneachofthe

roducedin~ 61toreducethe( n+ 1 fo ld

upiedby thefluid tosimplecontinuity.

anypointof thefluidwillthenbe a

6 0 ( x ) e u a l to t h e su m o f th e s ep a ra t e



s s_ u s e # p d



so lutionb= tan- ly / consideredin~ 56

ti o n V 2 0 = 0 a n d ob i o us l y sa t is f ie s t he

merelyfortheannularspacewithrectangular

onsidered butforthehollowspace

olutionobtainedbycarryingaclosed

n d an y a i s ( O Z ) n o t cu t ti n g th e c ur e

w esha ll infuturecallaho llow circularring.

onpossiblewithinafi edhollow

ev elocitypotentialisproportionalto

ridianplanethroughanypoint anda

so lidangle a subtendedatanypoint

y a n in f in i te s im a l pl a ne a r ea A i n a ny f i e d p os i ti o n

tionV 2a=0. Thisw ell- now nproposit ion

ingA attheorigin andperpendicularto

e

A - . . .. .. . .. 5 ,

3 d ( x 2 + 2 y+ 2 ( 5

erif ied.

at( x , y z ) byanysingleclosed

subtendedatthe samepointbyall

di ideany limitedsurfaceha ingthis

edge. [ C onsiderparticularly cur essuch

efirst threediagramsof~ 58. Hence

lesubtendedat( x , y z ) by thissurface

20isfulf il led. Hence j representsthev e locity

motionpossiblefora li uidcontained

edv esse l w ithinw hichisf i ed atan

uterboundingsurface aninfinitely

mof theclosedcur e inq uestion.

e ampleforwhichthecur eisa

hesimplestspecimenof cyclicirrotational

sthato fE ample(1 is toaseto fpara lle l

entialbeingtheapparentareaofa

eaofaspherica le ll ipse isreadily found

siblereadilyintermsofa completeelliptic

andthereforeintermsofincomplete

andsecondclasses.Thee ui-potential

eablebyaidof Legendre stables.B ut



s s_ u s e # p d



T I O N

w eowetheremar ableandusefuldisco ery

estreamlines( orlinesperpendicularto

aces aree pressible intermsofcomplete

econdclasses.Theyarethereforeeasily

re stables. Theanne eddiagram of

uchuse later showsthesecur esasca lculatedanddraw nbyMrMacfarlanef romHelmholt ' sformula

ctangularco-ordinates. A nimpro ed

escribed inanoteby MrCler

ehask indlya llowedmetoappendtothis

c it y a nd M a gn e ti s m v o l . i â € ”

^ f e Z - Z - y 0



s s_ u s e # p d



h e mo t io n d es c ri b ed i n E a mp l e ( 2 w i ll

eanysolidringformedby solidifyingand

ofthefluidboundedby streamlines

hinwire.Thusweha easolidthic

garing oranendlessk notasil lustrated

of~ 59 o fpeculiarsectiona lfigure

nesroundthe arbitrarycur eof

dthecyclicirro tationa lmotionw hich ifplaced

rmits isthatw hoseve locitypotentia lis

gledefinedgeometricallyin thegeneral

a m pl e ( 2 .

mpoundedacyclicandpolycyclicirrotationalmotion- inetico-statics.Thewor doneintheoperation

lateddirectlyby summingtheproducts

nitesimalareaofthesurface intothe

uidcontiguouswith thisareamo es

mal fora llpartso f thesurface w hether

r wherethegeneticpressureis applied

isionsofthewholetimefromthe

tion.

w or done andfdttime- integration

onupto anyinstant.Atanypre ious

ure q theve locity andbtheve locity

ntiguousto anyelementdoofthe

thedifferenceoffluidpressuresonthetwo

9 o foneof the interna lbarriers andNthe

uidv elocitycontiguoustoeitherdoor

statemente pressedinsymbolsis

o + I f k N d s . .. . .. . .. . .. . .. . .. ( 6 ,

these eralbarriersifthereare more

generalhydro inetictheoremfor

5 9 ( 6 c o mp a re w i th ~ 3 1 ( 5 ] , w i th b e p r es s ed

tsofapo intmo ingw iththef luid w e

7

epressuretobe impulsi e sothatthere is

apeeither oftheboundingsurfaceorof



s s_ u s e # p d



T I O N

fdt.Thiswillalsoimply thatdo/dt

arisonw ithqq 2 sothat

ationof~ 57w eha e

9 .

reachbarriersurface.

f f Ad . .. .. .. .. 1 0 .

ngmotionof theboundingsurfaceand

rs maybev ariedq uitearbitrarilyfrom

ftheimpulse sothatthehistory

c uisitionoftheprescribedfinal

therdif ferent andnote ensimultaneous

oundingsurface. Thusk , andk 2 may

ionsof t pro idedonly f ldtandf 2dt

a lues w hichwesha lldenoteby f ltandt2

e a m pl e w e m ay s u pp o se f t o h a e a t e a c h

reoneandthesameproportionofits final

at t er b e d en o te d b y D a n d if w e p ut . = W . . ( 1 1 ,

natesofposition butmayofcoursebe

hetime. Hence obser ingthat

s 1 ( 1 0 b e co m es

f f bd s . .. . .. . .. . .. 1 2 .

mberof thise uationdoubledagreesw ith

m em be r s of ( 7 ~ 5 7 w it h f a nd l e a ch

ndthef irstmemberof thate uationbecomes

yof thew holemotion. Hence w hen

0 ( 7 o f ~ 5 7 e p re ss es t he e u at io n of e ne rg y



s s_ u s e # p d



ration o f thef luidmotioncorrespondingto

bypressuresvary ingthroughoutaccording

etime thefirstmemberbeingtwice

emotiongenerated andthesecondtwice

ocess.

ample le tussupposethe initia ting

asfirsttogeneratea motioncorrespondingtov elocitypotentialb andafterthattochange the

btob+ b , denotingbybandO ' any tw o

0 = D andeachfulf i ll ingLaplace s

gmentationf romzeroto( , andagain

uniformthroughthew holef luid. Thew or

f o un d a s ab o e ( 1 2 ,

b ds ] . .. .. .. .. .. .. .. 1 ) ,

c . d en ot e th e cy cl ic c on st an ts r el at i e t o f a s fR k 2 & a m p c .

andtheaddit ionalw or doneinthesecondprocess

bp ) d o+ K ' f f( 2 ib + b e ) d A . .. .. .( 1 4 .

a e s e en ( ~ 6 ) t h at t h e ac t ua l f lu i d mo t io n

hollyonthe normalv elocityateach

ceand thev aluesofthecyclic constants

doneingeneratingitoughttobe independentof theorderandlaw of theac uisit ionofv e locityatthe

of theatta inmentof theva luesof these era l

t h e su m o f ( 1 ) a n d ( 1 4 o u gh t t o be

t i ff or 4 i n ( 1 2 w e su bs ti tu te b + c , t he

a lueandthatof thesumof ( 1 ) and(14

b do + , ( S c J B f ' d s- ~ ' f fb ds ] . .. 15 ;

iff erencebetweenthetwoe ualsecond

57forthecaseof

0

ce thee ua lityo f thesecondmembersof

utestheana lytica lreconcilia t ionof thee uations

esofgenerationofthe samefluid



s s_ u s e # p d



Y V E L O C I T Y O F A C I RC U L A R V O R TE X

t stranslationofHelmholt ' s2emoironV orte

x III. 1867 511-512.

a rl y a s ma y b e He l mh o lt ' s n o ta t io n l e t g be

a iso fauniformv orte - ring andathe

score( w hichwillbeappro imate ly

comparisonw ithg , thevorte motion

re isnomolecularrotationin anypart

hiscore andthatinthecoretheangular

arrotationisappro imatelyw orrigorously

stanceX f romthestra ighta is.

f translationisappro imatelye ua lto

meorderasthismultipliedbya/ gbeing

uidatthesurface ofthecoreis appro imatelyconstantande ualtocoa.At thecentreofthering itis

andW respecti e ly andifTbethe

w ethereforeha e



s s_ u s e # p d



NA MIC S [ 3

nslationisv erylargeincomparison

longthea isthroughthecentreof the

sso smallthatlog8g/ais largeincormparisonwith27r.B utthev elocityoftranslationisalwayssmall

elocityofthefluidat thesurfaceofthe

thesmalleristhediameterof thesectionin

eterofthering.

mpletelythedifficultywhichhas

rencetothe translationofinfinitely

Iha eonlysucceededinobta iningthem

fmymathematicalpaper( April29

ietyofEdinburgh buthopetobea llowed

hatpapershould itbeacceptedfor the



s s_ u s e # p d



C SO L U T I O N S A ND

.

ca l M ag a i n e V o l . X L I I. N o . 1 8 71 p p . 3 6 2 - 7 7

Lectures 1904 pp. 584-601.

F F R E E SO L I D S T HR O U G H A L IQ U I D .

producedbyaf ree lymo ingSo lid.

ththefollowinge tractfromthe

u r n al o f d at e J a n ua r y 6 1 8 5 8: L e t X , | 9 Z , X , t 1 j 2 b e r ec t an g ul a r co mp o ne n ts o f a n

mpulsi ecoupleappliedtoaso lidof in ariableshape w ithorw ithoutinertiao f itsow n inaperfect

, w a p a r b e th e c om p on e nt s o f li n ea r a nd

ated. Then if thev isv i at( tw icethe

thew holemotionbe asitcannotbutbe

io n

2 + [ v , ] v 2 + ... + 2[ v , u v + 2 [ w u wu + ...

. ..

[ v , v ] , & amp c. denote21constantcoef f icientsdeterminableby transcendenta lana ly sisf romtheformof thesurface

n o l ingonlyell iptictranscendenta lsw hen

n o l ing o fcourse themomentsof

emustha e

, t v + [ W , n w + [ u u + [ p u p + [ p ] a - = , & amp c .

[ v , W ] v + [ w U ] w + [ W , W ] W + [ p = ] p + [ a- i a- = t & amp c .

eProceedingsoftheRoyalSocietyof Edinburghfor

.fromletterstoProfessorTait ofAugust1871.PartV .

cationSeptember1871.

ad o f I Q , i s u se d t o de n ot e t he " m ec h an i ca l v a l u e " o r a s

k ineticenergy" o f themotion.



s s_ u s e # p d



C SO L U T I O N S A ND O B S E RV A T IO N S [ 4

, Y Z , andacontinuouscouple

t o a e s f i e d i n th e b od y i s a pp l ie d a n d if

d en o te t h e im p ul s i e f o rc e a nd c o up l e ca p ab l e of g e ne r at i ng f r om r e st t h e mo t io n u v , w w r p - w hi c h e i s ts i n

merelymathematica lly ifX & amp c. denote

glinearfunctionsof thecomponentsof

ns o f m ot i on a r e as f o ll o ws : dX _ 3 + p = X , d = & a m p c .

d t

mA O - + J a p= L I

p + . Mw = M

hen

Z = 0 L= O , M = O , N = O ... 2 ,

ob iouslyare

c o ns t .. .. . .. . .. . .. . .. . .. . .. 3 ) ,

stant

- = c o ns t .. .. . .. . .. . .. . . 4 ,

mentumconstant and

+ I + p tM + o - = Q . .. .. .. .. 5 .

ommunicatedinaletterto Professor

robably J anuary 1858 andtheyw erereferred

inhisf irstpaperonStream- lines

a lSocietyofLondon , J uly186 .

tedtotheRoyalSocietyof Edinburgh andthefollowingproofisadded: Thesee uationswillbev erycon enientlycalledtheEuleriane uationsof

ondprecise ly toEuler se uationsfortherotationofa

dethemasaparticularcase. A sEulerseemstoha ebeen

tionsofmotionintermsofco-ordinatecomponentsofve locity

f i edre lati e ly tothemo ingbody itw il lbenotonly

st todesignateas" Euleriane uations" anye uationsofmotion

ence w hetherforposit ion orv e locity ormomentof

orcouple mo ew iththebodyorthebodiesw hosemotion

noteboo o fdate1858 containingthisearly statementof the

asbeenpreser ed. F orde e lopmentsseeLamb s



s s_ u s e # p d



SO L U T I O N S A ND O B S E RV A T IO N S

hisin estigationwasto il lustratedynamica le f fectsofhe ligo idalproperty ( thatis rightorlef t-handed

ofcompleteisotropy withheli9oidal

chthecoef f icientsinthequadratice pression

nditions.

v ] = [ w w ( l et m be th ei rc om mo nv a l ue

= [ o- P ] , , n , , ,

p = [ w , ] , , h , ,

, U ] = [ , v ] = 0 [ p a = [ ] , P = [ o p = 0o

- = [ v , - = [ , C = [ W , = ] = [ w p = 0

2+ w2 + n( W2 + p2 + a2 + 2h( ur+ v p+ wo } ( 11 .

ore theEuleriane uations( 1 become

- wp = X , & a mp c .

= L & a mp c .. 1 1 .

f referencef i edre lati e ly

remainsunchangedw henthe linesof

yother threelinesatrightangles to

i t i s ea s il y s ho wn d i re c tl y f ro m ( 6 ( 7 ,

te r in g t he n o ta t io n w e t a e u v , w t o de n ot e

locityofPparalle lto threef i edrectangularlines andA p athecomponentsof thebody sangular

nes w eha e

amp c .

= L & a mp c .

f referencef i edinspace

ientthantheEuleriane uations. . .. 12 ,

uations whenneitherforcenor

X = 0 & a m p c . L = 0 & a m p c . , p r e se n ts n o

isreadily seenf rom~ 21( " V orte



s s_ u s e # p d



N O F F R E E S O L I DS T H RO U G H A L IQ U I D 7

w henthe impulse isbothtranslatoryand

roundwhichthebody isisotropic

rcleorspira lsoastok eepataconstant

iso f the impulse " andthatthecomponentsofangularv e locity roundthethreef i edrectangulara es

e madebyattachingpro ecting

aglobe inproperposit ions forinstance

atthemiddlesof thetwel eq uadrantsof

idingtheglobe intoe ightquadranta l

theglobeandthevanesof lightpaper a

ghand lightenoughtoillustrateby

motionsofanisotropicheli oid

e li uid. B utcuriousphenomena not

ntin estigation w ill nodoubt onaccount

r e d .

roughaperforatedSo lid. -7/

mo eablerigidbody inf inite ly

ofotherrigidbodies f i edormo eable

raperturesthroughit andlettherebe

circulations( . 60 " V orte Motion )

rbethecomponentsof the" impulse

r ee f i e d a e s a n d X , / t v i t s mo m en t s

abo e w itha llnotationthesame w estil l

o r te M ot io n" )

t d .

a mp c .

aquadraticfunctionof thecomponentsof

e n ow h a e

] U + ...+ 2 u v ] v +... ...... 1 ) ,

ergyofthefluid motionwhenthesolid

u u U 2 + . . . i s t h e s am e q u a d ra t ic a s b ef o re .

, [ u v ] , & amp c.aredeterminablebyatranscendentalanalysis ofwhichthecharacteris notatallinfluenced



s s_ u s e # p d



[ 5

I N D A ND C A PI L LA R IT Y O N W A V E S

DF R IC T IONLESS.

indonW a esinw atersupposed

rofessorTait o fdateA ugust16 1871.

allydownwardsand0Yhori ontal let

at . . . .. . . .. . . .. . . .. . . . 1

ectionof thew aterbyaplaneperpendiculartothew a e- ridges andleth( theha lfw a e-height be

sonw ith2rr/ n( thew a e- length . The

elocityof thew ateratthesurface isthen

. . . .. . . .. . . .. . . .. . 2 ;

nf initesimal mustbethev a lueofdl/ d

if4 denotetheve locity -potentia la tanypoint

te r . No w b ec a us e

nofy andafunctionofx w hich

= oo itmustbeof theform

ndentofx andy . Hence ta ingdo/d ,

nd e u at in g it t o ( 2 , w e ha e

n ah c os ( n y -n at ;

a n d e = n a t s o t ha t w eh a e

a t . .. . .. . .. . .. . .. . .. 3 ) .

ed resultssimplyfromtheassumptions

ss thatithasbeenatrest andthat

themannerspecif iedby ( 1 .



s s_ u s e # p d




nowind w henthew a e- lengthis27r/n.

.. . . .. . . .. . . .. . . .. 14 ,

2 ( 1 4 ,

eve locityof thesamew a esw henthere

, inthedirectionofpropagationof thew a es.

w e ha e

& g t } @ . 15 .

owingconclusions:

& lt w V / 1 + ) / /-

t i e a n d ne g at i e t h at i s t o sa y w a e s

stthew ind. Thepositi ev a lue isa lways

w a estra e lfasterw iththanaga instthe

a estra e ll ingaga instthew indisa lw ays

ityw ithoutw ind.

l t 2 w t he v e l oc it y of w a e s t ra e ll in g wi th t he

henV = 2w thev e locityof thew a es

bedby thew ind aresultob iousw ithout

isincorrect andiscorrectedinthereprintinB a lt imore

u lt s ( 1 , ( 2 , ( 3 ) , ( 4 a r e re p la c ed b y t he f o ll o wi n g:

n gt h 2 7 r/ n t h e g r ea t es t w a e - e l oc i ty i s w ^ / 1 + ) , w hi c h is

elocityofthewind.It isinterestingtoseethat with

hanthatof thewa es andinthedirectionof thew a es

nstance thew a e-speedw ithnowind w hichisw isless

ofw t ( orabout1/ 1650 ) thanthespeedwhenthew indisw ith

peed. Thee planationclearly isthatw hentheairismotionlessrelati e ly tothew a ecrestsandho llowsitsinertia isnotca lledintoplay.

- 28 7x

sz ero thatistosay staticcorrugationsofw a e- length

ibratedbywindofve locity

ouldbeunstable.

= 1 + .2 8- 7 1 + 8

e ual.

- & gt

aginary andthereforethewindwouldblowintospin-drift

or s h or t er . T h en f o ll o ws ( 1 6 a n d ( 1 6 ) .



s s_ u s e # p d



O F W I N D A N D SU R A C E- T EN S IO N O N W A V E S 7 9

t 2 w t h e v e l oc i ty o f wa e s t ra e l li n g wi t h th e

ocityof thesamew a es w ithoutw ind.

g t w . 1 + - / a- w a e s o f su ch l en gt h th at w wo ul d

utwind cannottra e laga instthew ind.

t w ( 1 + o / / o t he re c an no t be w a e s o f so s ma ll

heundisturbedvelocity isw andthe

risessentia llyunstable. A nd( 1 ) shows

eofw is

.......

ane le e lsurface isunstable if the

ceeds

1 6 ) .

fessorTa it o fdateA ugust2 , 1871.

a eonw aterwhose length

t wh er e

.. .. . .. . .. 1 ,

ua lwaysseeane q uisitepatternof ripplesin

thesurfaceofw aterandmo inghori onta llyatanyspeed fastorslow . Theripple -lengthisthe

tion

18 ,

o f theso lid. The lattermaybeasa il ing esse lorarow -boat apo lehe ldv ertica llyandcarriedhori onta lly ani orypencil- case apen nife -bladee itheredgeor

best a f ishing- linek eptappro imate ly

thangingdownbelowwater whilecarried

perhourbya becalmedv essel.The

otsadmirably ripplesinf ront and

ity ( X thegreaterrooto f samee uation

tobebeca lmedagain Isha lltry

olepattern showingthetransition

ow a es. Whenthespeedw ithw hich

tablee enif thea irw eref rictionless. B . L. reprint.

r= l7centim. ( seePartV . .



s s_ u s e # p d



I N D O N W A V E S

gharea lf luidsuchasa irorw ater.

esandhollow sofaf i edso lid(such

must becauseof thev iscosityof the

erforceontheslopesfacingitthanonthe

aregularseriesofw a esatseaconsistedofaso lidbodymo ingw iththeactua lve locityof thew a es

uponit o ritw oulddow or uponthe

ve locityof thew indw eregreaterorlessthan

es. Thiscasedoesnotaf fordane act

w indonw a es becausethesurface

mo eforw ardw iththev e locityof the

urrow edsoliddo. Stil l itmaybee pected

of thew inde ceedsthev e locityofpropagationof thew a es therewillbeagreaterpressureonthe

heanteriorslopesof thewa es and

at w h en t h e v e l o ci t y of t h e wa e s e c e ed s t he

orisinthedirectionoppositetothato f the

eaterpressureontheanteriorthanon

wa es.Inthefirst casethetendency

e inthesecondcasetodiminishit.

seriesofwa esofacertain height

certainforceof windorgradually

otbeing strongenoughtosustain

dof fhand. Tow ardsansw eringitSto es s

r aga instv iscosityofw aterre uiredto

gi esamostimportantandsuggesti e insta lment. B utnotheoretica lso lution andvery litt leo fe perimental

re ferredtow ithrespecttotheeddyingsof

etopsof thew a es tow hich by its

essureontheposteriorthanonthe

uenceofthewindin sustainingandmaintainingwa esischieflyif notaltogetherdue.

encalledthreedaysago byMrF roude

r t on W a e s ( B r i ti s h As s oc i at i on Y o r ,

emar able il lustrationorindicationof the

f theinfluenceofwindon wa es

w indmuste ceedthatof thew a es in

Lethim[ anobser erstudyingthe

C ambridgePhilosophica lSociety 1851( Ef fecto f Interna l

theMotionofPendulums " SectionV . .

6



s s_ u s e # p d




e duringthesuccessi estagesofan

aca lmtoastorm beginhisobser ations

nthesurfaceofthewateris smoothand

imagesofsurroundingob ects. This

ctedbye enaslightmotionof the

lessthanha lfamileanhour( 81in. per

isturbthe smoothnessofthereflecting

rflittingalongthesurfacefrompoint

edtodestroy theperfectionof the

ndondeparting thesurfaceremains

ea irha eav elocityofaboutamilean

waterbecomesless capableofdistinct

ser ingit insuchacondit ion it istobe

nofthis reflectingpowerisowing

minutecorrugationsofthesuperficial

fthethird order.Thesecorrugations

thewateraneffectv erysimilarto

glasswhichweseecorrugatedfor

heirtransparency andthesecorrugationsatoncepre enttheeyefromdistinguishingformsat a

ddiminishtheperfectionofformsreflected

his appearanceiswellk nownas

hwhichthefish seetheircaptors.

cehas thisdistinguishingcircumstance

esurfaceceasealmost simultaneously

disturbing cause sothataspot

edirectactionof thewindremains

f thethirdorderbeingincapableof tra e ll ing

siderabledistance e ceptwhenunder

original disturbingforce. This

f presentforce notofthatwhich

itgi esthatdeepblac nesstothe

ccustomedtoregardasaninde ofthe

oftenastheforerunnerofmore.

o fw a emotionistobeobser edw hen

cting onthesmoothwaterhasincreased

llwa esthenbegintorise uniformly

of thew ater thesearew a esof the

erthew aterw ithconsiderableregularity .

earf romtheridgesof thesew a es but



s s_ u s e # p d



I N D O N W A V E S

thehollowsbetweenthem andon

sew a es. Theregularityof thedistributionof thesesecondarywa eso erthesurface isremar able

ch ofamplitude andacoupleof

geasthev e locityordurationof thewa e

ontermina lw a esunite theridgesincrease

hewa esbecomecusped andare

ondorder. Theycontinueenlargingtheir

pthtowhich theyproducetheagitation

ywiththeirmagnitude thesurface

co eredw ithw a esofnearlyuniform

or" wa esofthethird order referred

inignoranceofhisobser ationsonthis

hadca lled" ripples. Theve locityof8- inches

ersecondisprecise ly theve locityhehad

yhisobser ations fortheve locityof

t-ridgedwa esstreamingobli uely

pathofasmallbodymo ingatspeeds

persecond anditagreesremar ably

perimentaldeterminationofthe

e- e locity ( 2 centimetrespersecond

asnote plicitlypo intedoutthathis

nchespersecondwasanabsoluteminimum

. B utthe ideaofaminimumvelocityof

ebeenfarf romhismindw henhef i ed

astheminimumofwindthat can

toappearinNatureon the26th

g i e n e t r ac t s fr o m Ru s se l l s R e po r t

uotationw hichhegi esf romPonce letand

f theF renchInstitutefor1829 showing

sonrippleshadbeen anticipated.Ineed

theseanticipationsdonotinclude

amicaltheorywhichIha egi en and

ew tomew henPartsIII IV andV o f

nwerewritten.



s s_ u s e # p d




e s u nd e r mo t i e p o we r o f Gr a i t y an d C oh e si o n

.

of w i nd c o ns i de r ( 1 ) a n d in t ro d uc e

1 7 i n i t. I t b ec o me s

. .. . . .. 19 .

ue

tothe caseofairandwater we

ebetw eengandg astheva lueofa is

nTandT , a lthoughit istoberemar ed

Tthatisordinarily ca lculatedf rom

ryattraction. F rome perimentsofGayLussac sitappearsthattheva lueofT isabout' 07 o fagramme

thatistosay intermsof thek inetic

grammeasunitofmass

aterunity ( asthatofthe lowerli uid

w emustta eonecentimetreasunito f length.

dasunito f t ime w eha e

n

ncentimetrespersecond corresponding

h e n l/ n = / 0 7 = 2 7 ( t h at i s w h en t h e

ntimetre , thev e locityhasaminimumvalue

cond.

theorywhichrelatestotheeffectof

uidsoccurredtomeinconse uenceof

edasetofveryshortw a esad ancing

ontofabodymo ingslowly throughw ater



s s_ u s e # p d



E S .

l . v . 1 87 1 p p. 1 - .

eredcohesionofwater( capillary

w hichw ouldseriouslydisturbsuche perimentsasyouw erema ing if ontoosmallasca le. Parto f its

hewa esgeneratedbytowingyour

. Iha eof tenhadinmymindthe

af fectedbygra ityandcohesionj o intly but

ringittoan issuebyacuriousphenomenonwhichwenoticed atthesurfaceofthe waterrounda

ngoutofOban( beca lmed atabout

hthewater.Thespeedwasso small

inea lmostv erticallydownw ards sothat

gementwasmerelyathinstraightrod

andmo edthroughsmoothw ateratspeeds

three- uartersofamileperhour. I

rs andotherformsofmo ingso lids butthey

eof them sogoodaresultasthef ishing- line.

shing-lineseemedto fa ourtheresult

tsroughnessinterferedmuchwithit. Isha ll

theropportunityof try ingasmoothroundrod

ertica lbya leadw eighthangingdow nunder

w hile it isheldupby theotherend. The

w ithoutanyotherappliancepro edamply

ygoodresults.

asane tremely f ineandnumerous

cedingthesolid muchlongerw a esfo llow ing

obli uew a esstreamingoff intheusual

oneachside intow hichthew a esin

ertoMrW. F roude bySirW . Thomson.



s s_ u s e # p d



herearmergedsoasto formabeautiful

thetacticsofw hichIha enotbeen

therto.Thediameterofthe " solid"

g- line be ingonly tw oorthreemill imetres andthe longesto f theobser edw a esf i eorsi centimetres it isclearthatthew a esatdistancesinanydirections

gf if teenortw entycentimetres w ere

stosaymo ingeachasif itw erepart

formparallelwa esundisturbedbyany

esseenrightinf rontandrightinrear

mmediatelyanob iousresultoftheory

thswiththesamev elocityofpropagation.

fallingoff thewa esinrearof thefishinglinebecameshorterandthosein ad ancelonger showinganother

. Thespeedfurtherdim inishing oneset

heotherlengthen untiltheybecome as

h o f thesamelengths andtheobli ue

ter eningpatternopenouttoanobtuse

angles. F orav eryshortt imeasetof

eforeandsomebehindthef ishing- line and

hthesamev elocity w ereseen. Thespeed

ternofwa esdisappearedaltogether.

fishing-line( producedfore ampleby

er causedcircularringsofw a estodi erge

nf rontad ancingatagreaterspeed

an thatofthefishing-line.All these

eryremar ablyageometryofripples

nyyearsagotothe Philosophical

w hi c h h o we e r s o f ar a s I c an r e co l le c t

ectwerenotdiscussed.Thespeedof

stheuniformsystemofpara lle lwa esbefore

rlyanabso luteminimumw a e- e locity

tytowhichthecommonv elocityofthe

shorterw a esinf rontwasreducedby

engtheningthelatterto ane uality

meweightper centimetreofbreadth

ofaw atersurface(ca lculatedf rom

ssac containedinPoisson stheoryof

rpurewateratatemperature so faras



s s_ u s e # p d



C ent. andonegrammeasthemassofa

fortheminimumvelocityofpropagation

centimetrespersecond . Theminimum

aw atermaybee pectedtobenotv ery

ouldofcoursebe thesameifthe

waterweregreater thanthatofpure

eratio asthedensity.

r be ingbeca lmedintheSoundof

entopportunity w iththeassistanceof

mybrotherf romB elfast o fdeterminingby

mw a ev elocityw ithsomeapproachto

washungata distanceoftwoorthree

sside soastocutthewateratapo intnot

motionofthe v essel.Thespeedwas

totheseapieces ofpaperpre iously

ingthe irt imesof transitacrosspara lle lplanes

metresasunder f i edre lati e ly tothe

edec andgunwale . B yw atchingcarefully

dwa es w hichconnectedtheripplesin

rear Ihadseenthatit includedasetof

o f fobli ue lyoneachside andpresenting

edthemtobew a esof thecrit ica llength

umspeedofpropagation.Hencethe

efishing-lineperpendiculartothe fronts

etrueminimumvelocity . Tomeasure it

ecessarywasto measuretheanglebetween

esofridgesandhollows slopingaway

e andatthesametimetomeasure

hefishing-linewasdraggedthroughthe

suredbyholdingaj ointed' two-foot

hes asnearlyascouldbe j udged by the

sof linesofw a e- ridges. Theangleto

openedinthisad ustmentw asthe

itdow nonpaper draw ingtw ostra ight

andcompletingasimplegeometrical

properlyintroducedto representthe

hemo ingso lid there uiredminimum

prhour theonlyothermeasurementofve locity e ceptthe

ning w hichoughttobeusedinanypracticalmeasurement

second.



s s_ u s e # p d



adilyobta ined. Si obser ationsof thisk ind

w ow erere ectedasnotsatisfactory . The

the otherfour:V elocityof DeducedMinimum

a e - V e l oc i ty .

nd. 2 ' 0centimetrespersecond.

' 8 .

2 .

22 9.

ofthisresultto thetheoreticalestimate

econd w as o fcourse merelyaco incidence

hesi e forceofsea-w ateratthetemperature( notnoted o f theobser ationcannotbeverydif ferentf rom

f romGayLussac sobser ationsfor

hthetheoreticalformulaej ustnow

aperw hichIha ecommunicatedtothe

h andwhichwillprobablyappearsoon

ine. If2 centimetrespersecondbe

speed-theygi e1 7centimetresforthe

gth.

e toca llripples w a esof lengthsless

andgenera lly torestrictthenamew a es

ceedingit. If thisdist inctionisadopted

suchthat theshorterthelengthfrom

thevelocityofpropagation w hile for

engththegreaterthev e locityofpropagation. Themoti e forceof ripplesischie f ly cohesion thato fwa es

of lengthslessthanhalfa centimetre

isscarce ly sensible cohesionisnearly

eof ripplesisthesameasthatof the

d ofthesphericaltendencyofa drop

. Ina llw a esof lengthse ceedingf i e

theef fecto fcohesionispractica lly insensible

beregardedasw hollygra ity . This

echoiceyou ha emadeofdimensions

sconcernsescaping disturbancesdueto



s s_ u s e # p d



onintothe theoryofwa ese plains

been feltinconsideringthepatterns

the surfaceofwaterina finger-glass

a moistfingeronitslip. Ifnoother

ityw ereconcerned the lengthf romcrest

56 v ibrationspersecondwouldbea

erippleswould beq uiteundistinguishablewithouttheaidofa microscope andthedisturbanceofthe

i edasadimmingof thereflections

ngcohesionintoaccount If indthe length

pondingtotheperiodofI- ofasecond

a lengthwhichquitecorrespondstoordinary

ect.

ctedtheformulafortheperiod( P in

th( 1 thecohesi etensionof thesurface

o f th e f lu i d ( p , i s

edink ineticunits. F orw aterw eha e

ngtotheestimate Iha eta enf romPoisson

8 2 x - 0 74 = 7 . H e nc e f or w a te r

hanhalfacentimetretheerror from

islessthan5percent. o fP . When1

theerrorfromneglectingcohesionis less

eiod. It istoberemar edthat w hile

ngthtobe insensibletocohesion the

es uarerooto f the length forripples

sibletogra ity theperiodv ariesinthe

e length.

ledmyattentiontoMrScottR usse ll s

r i ti s h As s oc i at i on Y o r , 1 8 44 a s c on t ai n in g

of thephenomenawhichformedthesub ect

him If indinit undertheheading

f onegrammeink ineticunitsofforcecentimetresper



s s_ u s e # p d



O r d er o r " C ap il la ry W a e s a m os t

" ripples" ( asIha eca lledthem , seen

o inguniformly throughwater a lsoa

sellf romapaperofdate No . 16 1829

w here itseemsthisclassofw a esw as

afterpremisingthatthephenomenonis

yofa f inerodorbarislightlydippedina

adescriptionof thecur edseriesof ripples

yattentionin themannerdescribedinthe

ell sq uotationconcludesw ithastatement

efo llowing: -. . . ontrou eq uelesrides

ndlav itesseestmoyennementaudessous

to il lustratethislaw . SofarasIcan

ly longw a esfo llow inginrearof themo ing

cribedeitherby PonceletandLesbrosor

yshownintheplanconta inedinR ussell s

eshow nabo etheplan( ob iously intended

thewater-surfacebyav erticalplane

therearaswellastheripplesinf ront and

otescapedtheattentionof thatv eryacute

respecttothecur esof therippleridges R usse lldescribesthemasha ingtheappearanceofa

as whichseemsamorecorrect descriptionthanthatofPonceletandLesbros accordingtowhichthey

riesofparabo liccur es. It isclear

ydynamica ltheory thattheycannotbeaccurate

f ar a s I a m ye t a bl e t o j u d g e R u ss e ll s

misaverygoodrepresentationof the ir

hegeometricaldeterminationofa

bser ingtheanglebetweentheobli ue

esstreamingoutonthetwo sides

inches( 21~ centimetres persecond.

estimateof25centimetresper second

ofso lidrelati e ly to f luidw hichgi es

sse ll stermina lv e locityof211centimetres

mar ableharmonyw ithmytheoryand

chInstitute 1829.



s s_ u s e # p d



EX PER IENC EDB YSO LIDSIMMER SED

I D .

sof theR oya lSocietyofEdinburghfor1869-70

nsin[ inPapersonElectrostaticsand

p. 567-571.

[ ~ 60( z ) ] onceestablishedthrough

inamo ablesolidimmersedina li uid

withcirculationor circulationsunchanged

h ow e e r t h es ol id b e mo e d o r be nt a nd w ha te e r

f romotherbodies. Theso lid if rigidand

earlycontinueatrestre lati e ly tothef luid

edistance pro idedtherebenoother

ncefrom it.B utiftherebe any

withinanyfinite distancefromthe

a lforcesbetweenthem w hich ifnot

ationof force w illcausethemtomo e.

briumofrigidbodiesin thesecircumstancesmightbecalledK inetico-statics butitis inrealitya

simply . F orw ek now ofnocaseof true

tallofthe forcesarenotdueto motion

hehydrostaticsofgases than sto

w eperfectlyunderstandthecharactero f

thestaticsof l i u idsandelasticsolids w e

indofmolecularmotionisessentiallyconcerned.ThetheoremswhichI nowproposetobringbeforethe

eforcese periencedbybodiesmutually

hroughthemediationofamo ingli uid

remsofabstracthydro inetics areof

illustratingthegreatq uestionofthe

w ithoutfarthertit learetotheauthor spaperonV orte

shedintheTransactions( 1869 w hichconta insdef init ions

nthepresentarticle. Proofsofsuchofthe propositions

reproofareto befoundina continuationofthatpaper.



s s_ u s e # p d



sactionat adistanceareality oris

a ined asw enow belie emagneticand

byactionof inter eningmatter

siderf irstasingle f i edbodywithone

hit asaparticulare ample apieceof

end. Lettherebeirrotationalcirculationof thefluidthroughoneor moresuchapertures.Itis

~ 6 E a m .( 2 ] * t h a t th e v e l o ci t y of t h e

ghbourhoodagreesinmagnitudeand

telectro-magneticforce atthecorrespondingpoint intheneighbourhoodofanelectro-magnet

structedaccordingtothefollow ingspecif ication. The" core " onw hichthe" w ire isw ound istobeof

initediamagneticinducti ecapacity -

i eandshapeastheso lidimmersedin

maninfinitelythin layerorlayers

d eachaperture.Thewholestrength

rec onedinabso lutee lectro -magnetic

a lto thecirculationof thef luidthrough

/4-7r.Theresultantelectro-magnetic

umericallye ualtotheresultantfluid

ondingpointinthehydro ineticsystem

re ample theparticularcaseofa straight

letthediameterbeinfinitelysmall in

h. The" circulation w ille ceedby

uantitytheproductofthe v elocitywithin

ntheneighbourhoodofeachend at

comparisonwiththediameterofthe

sonwiththelength thestreamlines

tingfromtheend.Thev elocity

ndinw ardstow ardstheother w il lthereforebe in erse lyasthes uareof thedistancef romtheend.

bledistancesf romtheends thedis O rf romHelmholt ' sorigina lintegrationof thehydro inetice uations.

ncesare accordingtoF araday sv erye pressi e

to linesofmagneticforce w orseconductorsthana ir.

itediamagneticinducti ecapacityisa substance

flinesofmagneticforce orwhichisperfectlyimper ious



s s_ u s e # p d



PER IENC EDB YIMMERSEDSOLIDS

ywillbethe sameasthatofthe magnetic

dofan infinitelythinbarlongitudinally

mendtoend.

parisonbetweenfluidv elocityand

Euler sfancifultheoryofmagnetismis

Thiscomparison whichhasbeenlong

rrelationbetweenthemathematicaltheories

sm conductionofheat andhydro inetics

notdynamical. Whenw epass asw e

astrict lydynamica lcomparisonre lati e ly to

twohardsteelmagnets weshallfind

ctionbetweentw otubes w ithli uid

utw iththisremar abledifference that

nthetw ocases unli epo lesattracting

ginthemagneticsystem w hile inthe

tractionbetw eenli eendsandrepulsion

onsidertw oormoref i edbodies such

op.I.Themutualactionsoftwo of

butinoppositedirections tothose

gelectro-magnets. Theparticular

eshow sustheremar ableresult that

canha easystemofmutualaction in

ew ithforcev ary ingin erse lyasthes uare

f thee itendsof tubes openateachend

hem beplacedintheneighbourhood

eenteringendsbeatinfinitedistances the

besimply attractionsaccordingto

tubesonthissuppositionare

erefore asiseasilypro edf romtheconser ationofenergy thequantit iesf low ingoutperunito f t ime

ctedbythemutualinfluence.[ When

e relati epositionsoftwotubesby

diminutionofk ineticenergyof thef luidis

andatthesametimeanaugmentation

hee ternalspace. Theformerise ua l

e the latterise ua ltothew or done

icenergy fromthew hole li uidissimply

ne .



s s_ u s e # p d



e enifoneof thebodiesconsideredbemere lyaso lid w ithorw ithoutapertures ifw ith

circulationthroughthem. Insuchacaseas

gneticsystemconsistsofa magnetor

merelydiamagneticbody notitselfa

gthedistributionofmagneticforcearound

nce. Thus fore ample aspherica l

motionsurroundingaf i edbody

thereis cyclicirrotationalmotion

dpressurearesultantforce throughits

itetothate periencedbyasphereof

city similarlysituatedintheneighbourhoodofthec orrespondingelectro-magnet.Therefore according

the latter andthecomparisonassertedin

erienceaforcef romplacesof lesstow ards

locity irrespecti e lyof thedirectionof

ghbourhood aresulteasilydeduced

ulaforfluidpressurein hydro inetics.

atan elongateddiamagneticbody

tends astendsanelongatedferromagneticbody toplaceitslength alongthelinesofforce.Hence

onaf i eda isthroughitsmiddle inauniform

ndstoplace itslengthperpendicularlyacross

ak now nresult( ThomsonandTait s

3 5 . A ga in tw oglobesheldinauniform

iningthe ircentres re uire forcetopre ent

achingoneanother.Inthemagnetic

ofdiamagneticorferromagneticinducti e

rwhenheldin alineat rightangles

dro ineticresultsimilartothisfor

obes istobefoundinThomsonand

y ~ 3 3 2.

thebodyconsideredin~ III. [bean

and beactedonby forceappliedsoas

tantofthe fluidpressure calculated

II. f o rwhate erpositionitmay

ndif itbe influencedbesidesbyany

sorigina llypublished w ithoutlim itation isob iously fa lse

eonlyperce i edto-day . â € ” Sept. 1872.



s s_ u s e # p d



C EDB YIMMERSEDSOLIDS

cessuperimposedontheformer it

ouldmo eunderthe inf luenceof the

ne w erethef luidatrest e ceptinso

eby thebody sow nmotionthroughit.

opositionwasfirstpublishedmany

amesThomson onaccountofw hichhe

orte o f f reemobility to thecyclicirro tationalmotionsymmetrica lroundastraighta is. [ A ddit iona l

mepropositionholdsforaglobeof any

fluidmotionconsistingofcirculationor

f inerigidendlesscur eorcur esfor

dbody inthe li uid. Demonstrationto

ofthe RoyalSocietyofEdinburghfor

oya lSocietyofEdinburgh March4 1872[ inf ra p. 108 .



s s_ u s e # p d



N D RE P U L S IO N S D U E T O V I B R A T IO N .

etterstoProf . F . Guthrie f romthePhilosophica l

871 reprintedinPapersonElectrostaticsand

p . 57 1 -4 ~ ~ 7 4 1- .

1 4 th 1 8 70 .

dtheProceedingsof theRoya lSociety

n A pproachcausedbyV ibration "

reatinterest. Thee perimentsyou

beautifulillustrationsofthek nown

einabstracthydro inetics w ithw hich

piedinmathematica lin estigations

-motion.

eorem thea eragepressureatany

frictionlessfluidoriginallyat rest

ptinmotionbyso lidsmo ingtoand

anymanner thoughaf initespaceof

antdiminishedby theproductof the

areof thev e locity . Thisimmediate ly

demonstratedinyoure periments

rages uareofve locity isgreater

earestthetuning- for thanonthe

ouslythecardmustbe attractedby

oundittobe butit isnotsoeasyat

atthes uareof thea eragevelocity

rfacesof thetuning- for ne ttothe

ortionsofthev ibratingsurface.

ation howe er thattheattractionmust

oubtva lid asw emaycon inceourse l es

chbearsthe tuning-for andthe

omo ethroughthef luid. If the

dsthetuning-for andtherew ere



s s_ u s e # p d



N S A N D R EP U L S I O N S D U E T O V I B R A T IO N S 9 9

teforceon theremainderofthewhole

andsupport thew holesystemw ould

ndcontinuemo ingw ithanacce lerated

f theforceactingonthe card-an

t indeed bearguedthatthisresult

m ightbesa idthatthek ineticenergy

raduallytransformitselfintok inetic

mo ingthroughthef luid andof the

osingup behindthesolid.B ut

mostsuf f icestoputdow nsuchanargument

aticaltheory especiallythetheoryof

et i cs e p l ai n ed i n m ya r ti c le o n " V o r te m ot i on " n e ga t i e s i t.

tionwhichyouobser edagrees

agneticattractionin acertainideal

ecifiedby theapplicationofaprinciple

ic l e [ ~ 7 3 - 7 4 0 c o mm u ni c at e d to t h e

hinF ebruary last[ 1870 asanabstract

onofmypaperon" V orte -motion.

deal tuning-for twoglobesordis s

fo inthe line j o iningthe ircentres the

lbeabar withpolesofthesamename

leoppositepolein itsmiddle.Again

rdis isane ua landsim ilardiamagnetico fe tremediamagneticinducti ecapacity [ ~ 7 4 .

themagneticandthe diamagnetic

itetothecorrespondinghydro inetic

pplytheanalogy wemustsuppose

ary f romma imummagneti ationto

ane ua landoppositemagneti ationbac

miti emagneti ation andsoonperiodica lly . Theresultanto f f luidpressureonthedis isnotat

ppositetothe magneticforceatthe

butthea erageresultanto f thef luid

ea erageresultanto f themagneticforce .

thediamagneticisgenerallyrepulsion

e erthemagnetbeheld andisunaltered

sa lo f themagneti ation itfo llow sthat

oya lSocietyofEdinburgh read29thA pril 1867.



s s_ u s e # p d



thefluidpressure isanattractiononthe

- for intow hate erpositionthetuningfor beturnedre lati e ly to it. .. .

oubt curiouslycloseana logiesbetw een

sofmotionin contiguousfluidsof

thedistributionofmagneticforcein a

esofdifferentinducti ecapacities.

eoccupiedbyfrictionlessincompressible

ortionsthaninothers aso lidbesuddenly

ofthefluid motionfirstgeneratedagree

751-76 be low w iththepermanentlines

espondinglyheterogeneousmedium

ar-magnet tobesubstitutedforthe

cedw ithitsmagnetica isinthe lineof

oamounts thef luidv e locitymultiplied

e ualtotheresultantmagneticforce

articulardef init ion[ the" e lectromagnetic

Postscript ] o f theresultantmagneticforce

neousinducti ecapacity gi eninthe

bo e ~ 48ofmypaperonthe" Mathematica l

* beadopted. B utheretheanalogy

rtueofw hichasolidmo eable inaf luid

magneticinducti ecapacityk eeps

st[ contrast~ 751below inthehydro inetic

t ions J une21 1849. PublishedinPartI. fo r1851.



s s_ u s e # p d



. . . w e ha e ( H a mi l to n ia n f or m o f La g ra n ge s

. ( 1

' = K ' . ..

lek ineticenergyof thesystem andb

othesiso f I 7 . . . K ' . . . constant.

n g of X , K , K , X ' , . . . l e t B b e o ne

toberegardedgenera llyasmo able .

rsurfacef2across theapertureto

andconsiderthisbarrierasf i edre lati e ly

ormalcomponentve locity re lati e ly to

anypointo fQ; andletf fdadenote

oleareaof f2: then

. .. . . .. 2 ;

tffNdo-......................... 3 ) ,

pressionof thedef init ionofX . Tothe

withfl atanyinstant letpressurebe

eK perunito farea o erthew holearea

orce( orforceandcouple beappliedto

sitetotheresultanto f thispressuresupposed

rigidmateria lsurfacef2rigidlyconnectedw ithB . Themoti e( thatistosay systemof forces

K onthef luidsurface andforceand

ed constitutesthegenera lisedcomponent

[ T h om s on a n d Ta i t ~ 3 1 ( b ] ; f o r it

motionofB orotherbodiesof thesystem

andifX v arieswor isdoneattherate

orforcesthere maybeinthesystem.

sityof thef luidunity le tK denote

. M . ~ 6 0 ( a ] t o f t he f l ui d i n an y c ir c ui t c ro s si n g

otethetangentia lcomponentof theabsoluteve locityof the

cuit andfdslineintegrationonceround thecircuit.

hedbytheinitialsV .M.aretothe partalreadypublished

nV orte Motion. ( Transactionsof theR oya lSocietyof

1868-9.



s s_ u s e # p d



R ING-SHA PEDSOLIDS

nlyonce: it isthisw hichconstitutesthegenera lised

lati e ly toX [ ThomsonandTait ~ 3 1

M .~ 7 2 w eh a e

. .. . . .. . . .. 4 ,

est( orinanystateofmotionforw hich

b y t he m o ti e K d u ri n g ti m e t .

T is o fcourse necessarilyaq uadratic

dmomentum-components , I . . . K, K i . .

y functionsofEl b . . . butnecessarily

' , . . . Inconse uenceof thispeculiarity it is

-& amp c. -8:-/ / ~' -& amp c. ... + ( c K " ' , ... ......... 5 ,

tw oquadraticfunctions. Thisw emayclearly

th e n um b er o f t he v a r i a bl e s: 7 . .. a n d j t h e

thew holenumberofcoeff icientsinthesingle

p re ss in g r is ~ ( i + j ) ( i + + + 1 , w hi ch i s

mb e r of t h e co e ff i ci e nt s 2 i ( i + 1 + - ( j + 1

ctions togetherw iththe ij a a ilable

.. a , / , . .. .. .

uantit iesa a , . . . a . o . thusintroduced

memberthat

d.

' X d i= ' X % , ( 6 ^

5 , a n d us i ng t h es e w e fi n d

. .. . . .. . . ,

c. x = d - ' a - E- c . . . .. .. .. .. .. 8 .

w t h at - a + , - 3 4 - a + , & a m p c . a r e th e c on t ri b ut i on s t o th e f lu a c ro s s Q I 2 , & a m p c . g i e n b y th e s ep a ra t e v e l oc i ty T h e ge n er a l li m it a ti o n f o r im p ul s i e a c ti o n t h at t h e di s pl a ce m en t s ef f ec t ed

ll isnotnecessary inthiscase. Compare~ 5( 11 ,



s s_ u s e # p d



A nd( 7 show thattopre entthesolids

w henimpulsesK, K ' , . . . a reappliedtothe

races w emustapply tothemimpulses

ations

amp c r. 7 = / K + / + & a mp c . ... ... ( 9 .

so fmotion w eha e inthef irst

,

' , . . .. . .. . .. . .. . .. . .. ( 1 1 ;

lerationof ic underthe inf luenceofK,

celerationofa massundertheinfluence

hemotionsof theso lids , le t

- & amp c . q 7 0 = 7- / K - / -& a mp c . . ... 12 ;

a m p c . d e no t e v a r i at i on s o f Q o n t he h y po t he s is o f

nt.

, r e me m be r in g t ha t b IT / dr & a m p c . d en o te

he h y po t he s is o f I a 7 . . . f K c , . .. c o ns t an t

a d

d # s + c .

& a mp + - & amp

a m p c . C c. . + d ~ *

- & a mp & a mp c . + . ( 1 ) .

r

a .

+ & a mp C .

) +

, { d / , d / 9 b & gt

. - & a mp C .+ - . .. .. . ( 1 4 .

a c co r di n g to t h e no t at i on o f ( 1 2 , ~ 0 o . .. a r e

etsofthesolids duetotheirownmotion



s s_ u s e # p d



R ING-SHA PEDSOLIDS

otionof the li uid andthereforee lim inate: 7 . - . by ( 12 f rom( 14 . Thusw ef ind

& a mp

- + & a mp c .

a d \ ) + & amp c

- & a mp C .+ & a mp C . t- ( 1 5 ,

O d # r ~ ) dc y

po n di n g e u a ti o n fo r o & a m p c . a n d wi t h ( 1 1

m p c . a r e th e d es i re d e u a ti o ns o f m ot i on .

e of a p pl i ca t io n o f K , K ' , . . .( ~ 1 i s

yother( suchasthe inf luenceofgra ityona

mperaturesindif ferentparts isimpossible

hatistosay ahomogeneousincompressible

a e K = 0 K ' = 0 a n d fr o m ( 1 1

. . . a reconstants. [ Theyaresometimescalled

( V . M . ~ ~ 6 2 - 64 . T h e e u a ti o ns o f

omesimply

, c dd d ' 1

d oI d 1+ " ) * '

d y \ } + & a mp c _

ationsfor0 , 0 andw iththefo llow ing

b e tw ee n 0 o 7 0. . . an d d 4 . . .

( Q

0= , 0& amp c......... 17 .

-d + ) + & a m p c . b e d en ot ed b y{ ' , } . 1 8

. .. . .. . .. . .. . .. . .. . . 1 9 .

} , { 0 r , & amp c. linearfunctionsofthecyclic

entsdependingontheconfigurationofthe

enera lly regardedsimplyasgi enfunctionsof

, 0 . . . : andthee uationsofmotionare

+ & a mp 0 * 1 c & amp = b -t

d # ( 2

b . ..Q ( 2 0 .

[ I 0 } 1 O + o .& a mp c .=

0



s s_ u s e # p d



amiltonianform Q isregardedasa

q 0 4o . .. w ithitscoef f icientsfunctionsof

mp c. andDiappliedto it indicatesvaria tionsof thesecoeff icients. Ifnow weelim inate: , q 0 o . . . f romQ by the linear

1 7 i s a n ab b re i a te d e p r es s io n a n d so

s a q u a d ra t ic f u nc t io n o f r w 0 . .. w i th

f r q b 0 & a m p c . a n d if w e de n ot e b y

a m p c . v a r ia t io n s of Q d e pe n di n g on v a r ia t io n s of

by d Q / d ~ r d Q / l d & a m p c . v a r i a ti o ns o f Q

s of j , q t & a m p c . w e ha e [ c o mp a re T h om s on

) a n d ( 1 5 ]

21 ;

; Q d Q

dl

otionbecome

b

+ + d

0 B } ~ + . .. .- - -

d

of Lagrange sform withtheremar ableadditionofthetermsin ol ingthev elocitiessimply( in

icconstants dependingonthecyclic

sof thesecondmemberscontaintraces

ninthesymbolsb/ d b/ d+ , . . . .

thesee uations le t-= 0 = 0

s 0 = 0 o = 0 . . . a n d th e re f or e a ls o Q = 0

ot i on ( 1 6 ( n o w e u a t i on s o f e u i l i b ri u m

luenceofappliedforcesP ) , & amp c.

reduetothepo lycyclicmotionK, / C , . . . ,

p c = . .. .. .. .. . .. .. 2 ) ;

licationoftheprincipleof energy

augmentationofthewholeenergyproduced



s s_ u s e # p d



R ING-SHA PEDSOLIDS

ce m en t 8 r i s b / d . $ 8 f a n d T8 i s

ppliedforces . It ispro edin~~ 724-7 0ofav o lumeofco llectedpapersonelectricityandmagnetism

h at t Q / d - 4 b Q t / d f & a m p c . a r e th e c om p on e nt s

edbybodiesofperfectdiamagneticinducti e

neticfield analogous tothesupposed

Hencethemoti einfluenceofthe

idupontheso lidsine uil ibriumise ua l

agnetisminthemagneticanalogue.

epaperO ntheF orcese perienced

o ingLi uid w hichre latestothe

pthemo ableso lidsatrest. Thepresent

p.II.ofthatarticle tobefalse.Compare

orthecaseofasingleperforatedmo able

rs agreesubstantia llyw ithe uations( 6

mmunicationStotheR oya lSocietyofEdinburgh

0 0o . . . o f thepresentarticlecorrespond

& a m p c . o f t h e f or m er t h e , A . .. m e an t h e

tionsnowdemonstratedconstitutean

notreadilydisco eredorpro edby that

eprincipleofmomentum andmoment

cha lonewasfoundedthe in estigationof

a ly tica ldefinit ionofQ in~ 3 ( 5 ,

themo ablesolidsisperforated this

e ua ltothew holek ineticenergy ( E ,

onw ouldha ew eretherenomo able

eenergy ( W) w hichw ouldbegi enup

chonthissupposit ionf low sthroughthespace

so lids suddenly rigidif iedandbroughtto

. .. . . . (24 ,

independentoftheco-ordinatesofthe

mayput-W inplaceofQt inthee uations

cleon" TheF orcese periencedbySolidsimmersed

( P r oc e ed i ng s R . S. E . F e b ru a ry 1 8 70 r e pr i nt e d in V o l um e o f

e rs ~ ~ 7 3 - 7 40 .

Session1870-71 orreprintinPhilosophica lMlaga ine No ember1871.



s s_ u s e # p d



rthisslightmodif ication neednotbew ritten

ctlydefinedas thewholeq uantityof

o ethemo ablesolids eachtoaninf inite

idha ingaperforationwithcirculation

hthisdef init ion -W maybeputforQ lin

w ithoute clusionofcasesinw hichthere

uresin mo ablesolids.

ry simplecase thesub ectofmy

alSocietyoflastDecember inwhich

thoutproof . Lettherebeonlyonemo ing

lettheperforatedso lidorsolidsbereduced

ablerigidcur eorgroupofcur es

t h at i s e i th e r fi n it e a nd c l os e d o r i nf i ni t e , a n d

edso lid. Therigidcur eorcur esw ill

s asthe irpartissimply thato fcoresfor

otion. Inthiscase it iscon enientto

t he r ec ta ng ul ar c o- or di na te s ( x , y z ) o f th e

globe. Then becausethecores be ing

oobstructiontothemotionof the li uid

be m o i n g th r ou g h it w e ha e

) . . .. . .. . .. .. . .. . .. ( 2 5 ,

softhe globe togetherwithhalfthatof

nce

2 6 .

M

, 0 = y

onoccurs becauseinthepresentcase

a sw en ow ha e it a d + , d y+ y d , i sa

Topro ethis le tV betheve locitypotentia la tanypoint( a b c duetothemotionof theglobe

clicmotionof the li uid. Weha e

usof theglobe and

( y -b 2 + ( Z - c } 2.

theglobe a f teranymotionw hate er greatorsmall.

ninwhichithasbeenbefore the integra lquantityof

hascausedtocrossany f i edarea iszero .



s s_ u s e # p d



R ING-SHA PEDSOLIDS

mponentve locityof the li uidat( a b c

M u v ) w eh a e

F ( x , , z , , b + b c . ..... 27 ,

z , a b c = -r ( X -d + /d+ v ) D

b e a ny p o in t o f th e b ar r ie r s ur f ac e Q 2 ( ~ 2 , a n d

dr ec t io n c os i ne s o f th e n or m al . B y ( 2 o f ~ 2 w e s e e

themotionof theglobe isffNdr or

, y z , a b c do...... ( 28 .

a b c d o- = U ,

4 t ha t

... 29 .

on o f ~ 7 ( 1 8 f o r x , y . . . in s te a d of 4 & g t , . . .

, x = 0 { x , y = 0.

e e u a ti o ns o f m ot i on ( 2 2 , w i t h ( 2 4 , b e c om e

W d 2 W . 3 0 .

d y d z

essthattheglobemo esasamaterialparticle

r ce s ( X , Y Z ) e p r es s ly a p pl i ed t o i t w o ul d

ha ingW forpotential.

f courseeasily foundbya idofspherical

e locitypotentia l P o f thepo lycyclicmotion

thegloberemo ed andw hichw emust

w or ingitout( be low it isreadily

hypothetica lundisturbedmotion q denote

ointreally occupiedbythecentre

a e

............... 3 1 ,

a halftimesthev olumeoftheglobe

ticenergyofwhatwemaycall theinternal

pyingforaninstant intheundisturbed



s s_ u s e # p d



gidglobeinthe realsystem.Todefine

harmonicana ly sispro estheve locityof the

tationa llymo ingli uidglobetobe

e l oc i ty o f t he l i u i d at i t s ce n tr e a n d co n si d er

of the li uidsphere re lati e ly toarigid

e locityq . Thek ineticenergyof this

isdenotedbyw. R emar a lsothatif

itsparts the li uidglobew eresuddenly

yof thew holew ouldbee ua ltoq; and

r gi enupby therigidif iedglobeand

theglobeis suddenlybroughttorest

r re uiredtostarttheglobew ith

amotionlessli uid.

oc i ty p o te n ti a l at ( x , y z ) i n t he a c tu a l

entherigidglobe isf i ed. Letabethe

st a nc e o f ( x , y z ) f r om i t s ce n tr e a n d ff d o

ce.At anypointofthesurface ofthe

obe thecomponentv e locityperpendicular

ntheundisturbedmotionis(dP / dr . = , ;

epressureonthespherical surface

ve locitypotentia lo f thee ternall i u id

- - u n do e s an a m ou n t of w or e u a l to

mponentf romthatv a luetozero . O n

rnalv elocity-potentialisreducedfromP

undoneinthisprocessis

f ( P + ) d P. .. .. ( 3 2 .

P + Y d ' . . .. . .. . .. . .. . .. . . 3 2 .

elocity-potentialP+ s thereisnoflow

ricalsurface gi es

.. . 3 3 ) .

romtheproposition( ThomsonandTait s

496 thatany functionV , satisfy ingLaplace se uation

y2+ d2V / d 2throughoutasphericalspacehasforitsmeanv a lue

a lueatthecentre. F ordP / d satisf iesLaplace se uation.



s s_ u s e # p d



R ING-SHA PEDSOLIDS

p + & a mp )

P i - i - & a m p c .

.. .. .. 3 4 ,

+ & a mp C .

de e lopmentsofPandrre lati e ly to

beasorigin theformernecessarily

thelargestsphericalspacewhichcan be

s centrewithoutenclosinganypartof

cessarily con ergentthroughoutspace

B y ( 3 3 ) w e ha e

. 3 5 .

s

i P

Pi P i = 0

( 2 i + 1 f f d P. .. . .. . ( 3 6 .

taso lidspherica lharmonicof thef irstdegree

n of x , y z , p u t

C . .. .. .. .. .. .. .. .. ( 3 7 ,

A 2 + B 2 + C

2

B 2 + C G 2 . f f. d = q 2 x v o l u me o fg lo be = y 2 .

2 . 3 8 ;

om p ar i so n wi t h ( 3 1 ,

. 4 7 p+ . . .. .. .. .. 3 9

otheglobe isinf inite ly small

. .. . . .. . . .. . . .. . . ( 40 ,

halftimesthev olumeoftheglobule

elocityofthefluidin itsneighbourhood.

mulawhichI ga etwenty-fi eyears



s s_ u s e # p d



encedbyasmallsphere( whetherofferromagneticordiamagneticnon-crystallinesubstance inv irtue

cew hichite periencesinamagnetic

testra ightlineforthecoreasimple

ample isa f forded. Inthiscase theundisturbedmotionof thef luidisincirclesha ingtheircentresin

sw emaynow call it , andthe irplanesperpendicularto it. A sisw ellk now n thev e locityof irrotationa l

ghta isisin erse lyproportiona ltodistance

hepotentia lfunction W fortheforce

itesimalso lidsphere inthef luidisin erse ly

tanceof itscentref romthea is and

erse lyasthecubeof thedistance andis

o f thea is. Hence w hentheglobule

ndiculartothea is itdescribesoneor

esianspira lst. If itbepro ectedobli ue ly

onentv e locityparalle lto thea isw illremain

componentwillbe unaffectedbythatone

ftheglobuleonthe planeperpendicular

scribethesameCotesianspiral aswould

nomotionpara lle lto thea is. If the

nypositionitwill commencemo ing

ttractedbyaforcev ary ingin erse lyasthe

remar ablethatittra ersesatright

uidcurrentwithoutanyappliedforceto

asw emighterroneouslyatf irstsighte pect

yswiththeaugmentedstream.Aproperly

encewouldatoncepercei ethatthe

momentumroundthea isre uiresthe

y tow ardsit.

letobe ofthesamedensityas

nf inite ly small it ispro ectedinthe

sE periencedbySmallSpheresunderMagneticInf luence and

resentedbyDiamagneticSubstances ( Cambridgeand

urna l May1847 ; and" R emar sontheF orcese perienced

edF erromagneticorDiamagneticNon-crystallineSubstances ( Phil.lMag.O ctober1850 .ReprintofPapersonElectrostaticsand

4 - 66 8 M a cm i ll a n 1 8 72 .

amicsofaParticle ~ 149( 15 .



s s_ u s e # p d



R ING-SHA PEDSOLIDS

ocityof the li uid smotion itw il lmo e

mecirclew iththe li uid butthismotion

dtheneglectedtermw ( 3 9 addstothe

a itandStee le sDynamicsofaParticle

ec i es I V . c a se A = 0 a n d A B f i n i te a l so l i mi t in g

sI.andSpeciesV .Theglobulewill

theoppositedirectionifpro ectedwith

sitetothatofthe fluid.Iftheglobule

thedirectionof the li uid smotionor

e locity lessthanthatof the li uid it w il l

nspira l( SpeciesI. o fTa itandStee le ,

nitetime withaninfinitenumberof

dineitherof thosedirectionswitha

thanthatof the li uid itw il lmo ea long

eciesV . o fTa itandStee le , f romapseto

longtheasymptote ataninfinitedistance

ymptotef romthea isw illbe

+ )

anceof theapsef romthea is and/ c/27rct

uidatthatdistancef romthea is. If the

manypointin thedirectionofany

stdistancefromthea isisp itw il lbe

orescapef romit accordingasthecomponent

rpendiculartothea isislessorgreater

emar edthatine erycase inwhichthe

a is( e ceptthee tremeoneinw hich

itt le lessthanthatof thef luid andits

perpendiculartothe radiusv ector ,

oaches althoughithasalwaysan

o lutions iso f . f inite length andtherefore

entoreachthea isisf inite. C onsidering

aplaneperpendiculartothea is atany

mthea is le ttheglobulebepro ected

ngalinepassingatdistanceponeitherside

8



s s_ u s e # p d



C S.

gsof theR oyalSocietyofEdinburgh Session1875-76

Aug.1880.

r issteadymotionofv ortices.

o f " steadymotion. Themotionof

f luid orso lidandf luidmatterissa idtobe

tionremainse ua landsimilar andthe

sparticlese ua l how e ertheconf iguration

andhowe erdistantindi idua lmaterial

e fromthepointshomologoustotheir

ndnotsteadymotion: 1 A rigidbodysymmetricalroundana is settorotate

tscentreofgra ity andle ftf ree performs

odyha ingthreeune ualprincipa l

nyshape inaninf initehomogeneous

rmly roundany a lw aysthesame f i edline

ara lle lto thisline isacaseofsteady

igidbody inaninf inite li uidmo ingin

e(2 , andha ingcyclicirro tationalmotion

sperforations isacaseofsteadymotion.

rotationalmotionofli uidinthe

tionallymo ingportionoffluid ofthe

pro idedthedistributionof the

atthe shapeoftheportionendowed

.Theob ectofthepresentpaper

lconditionsforthefulfilmentofthis

estigate further thecondit ionsofstabil ity

motionsatisfyingtheconditionof



s s_ u s e # p d



ConditionforSteadinessofV orte

fluid smolecularrotationatany

stbethesameasif fortherotationa lly

luidw eresubstitutedaso lid w iththe

a iso f thef luid sactua lmolecular

edate erypo into f it andthew hole

scriptionswithit w erecompelledtomo e

gtothedescriptionofe ample( 2 . If

onof anymolecularrotation through

gdistributionoffluid- elocityaresuch

itwillbefulfilled throughalltime.

ditionforSteadinessofV orte

2 4 b e lo w v o r t i c it y a nd " i m pu l se g i e n

a imumoraminimum it isob ious

ysteady butstable . If w ithsame

sama imum-minimum themotionis

ybe eitherunstableorstable.

mholt ringisacaseofstable

nergyma imum-minimumforgi env orticity

rcularv orte ring w ithaninnerirrotationa lannularcore surroundedbyarotationa llymo ingannular

w ithirrotationa lcirculationoutsidea ll

ssteady if theouterandinner

therotationalshellareproperlyshaped

f theshe llbetoothin t. Inthiscase

mum-minimumforcirculargi env orticity

steadymotion the" resultantimpulse ( V . M. + ~ 8 isasimple impulsi e force w ithoutcouple :

dyofe ample( 3 ) isatoro id and

ionaland paralleltothea isofthe

snow w ell- now nfundamenta ltheoremsshowsthat f romn

e erypo into faninf inite f luid theve locityate erypo int

pressedsyntheticallybythesameformulaasthosefor

esultantforce o fapuree lectromagnet. ( Thomson s

ostaticsandMagnetism.

isde letedinacopyannotatedbyLordK el in.

personV orte MotionintheTransactionsof theRoya l

ethus referredtohenceforth.



s s_ u s e # p d



dingly interestingcasesofsteadymotion

chthat if appliedtoarigidbody it

cordingtoPo insot smethod toanimpulsi e

andacouplew iththislinefora is.

taindistributionsof v orticitygi ing

w iththic eningsandthinningsof the

esinonedirectionortheotherrounda

hwillbe in estigatedinafuturecommunication

suchcasesthecorrespondingrigid

2 hasbothrotationa landtranslationa l

suppose f irst thevorticity ( def ined

heforceresultanto f the impulsetobe

tionse pla inedbelow ~ 29 suchthatthe

mparisonwiththeaperture.Ta e

apieceofbloc t inpipe w ithitsends

rsw ell , bendit intoano a lform and

dedtwistroundthe longa iso f theo a l

stobenotinoneplane( f ig. 1 . A

llipseofthis

ectly determinate

heforceresultanto f

ationalmoment

~ 6 , a r e a ll g i e n

whatwe

modeofsingle

e motionw ith ig. .

ustratethesecondsteadymode commencew ithacircularringof f le iblew ire andpull itout at

neanother soastoma eit intoasit

nglew ithroundedcorners. Gi enow a

undtheradiustoeachcorner to theplane

athecorner and k eepingthecharacter

othew ire bendit intoacerta indeternlinateshapeproperforthedataof thevorte motion. Thisis

pp e i s s ub s ti t ut e d he r e fo r " v e r y st o ut l e ad w i re . ]

dsecond andthird andhighermodesofsteadymotion

oustothef irst second third andhigherfundamenta l

rator oro fastretchedcord oro f steadyundulatorymotion

al orinanendlesscha inofmutually repulsi e lin s.



s s_ u s e # p d



- core inthesecondsteadymodeofsingle

e motionwithrotationalmoment.The

edat by tw istingthecornersofa

dcorners thefourth by tw istingthecorners

ingroundedcorners thef if th by tw isting

on andsoon.

diagramsof toro idalhe licesacircle

j udgmentastotherelie fabo eand

eofthediagramwhich thecur erepresentedineachcasemust beimaginedtoha e.Thecirclemay

tobethecirculara iso fatoro ida lcore

besupposedtobew ound.

Iha esaid" gi earight-handed

eresult ineachcase asinf ig. 1 i l lustratesav orte motionforw hichthecorrespondingrigidbody

ices bya ll itsparticles roundthecentral

ow insteadof right-handedtw iststothe

hecornersof thetriangle s uare pentagon

e l e ft - ha n de d t wi s ts a s i n fi g s. 2 3 , 4 t h e re s ul t i n

motionforw hichthecorresponding

F i g . 4.

handedhelices.Itdepends ofcourse

edirectionsof theforceresultantand

ulse withnoambiguityinanycase

orms andinthelines ofmotionofthe

willberight-handedorleft-handed.

o fmotiontheenergy isama imumminimumforgi enforceresultantandgi encoupleresultanto f

essi e lydescribedabo earesuccessi e

m-minimumproblemof~ 4-adeterminate

so lutionsindicatedabo e butnoother



s s_ u s e # p d



rt icity isgi eninasinglesimpleringof the

motion forthecaseofav orte linew ithinf inite ly thincore bearsacloseanalogy tothefo llowing

em: indthecur ew hose lengthsha llbeaminimumw ithgi en

rea andgi enresultantarea lmoment

w ouldbe identica lw iththevorte problem

ey thinvorte - ringofgi env o lume

twerea functionsimplyofitsapertural

etricalproblemclearlyhasmultiple

selytothesolutionsofthe v orte

sofso lutionareclearlyverynearly

ms( infinitelyhighmodesidentical ,

e theso lutionderi edinthemannere pla inedabo e

ofN sides w henN isav erygreat

hate itherproblemmustleadtoaform

longregularspira lspringof theordinary

twoendsmeet andthenha ingitsends

dsoastogi eacontinuousendlessheli

teadof theordinarystra ightline-a is , and

ditscirculara is. Thiscur e Ica lla

se it l iesonatoroid , j ustasthecommon

culartoro idasimpleringgeneratedby there o lutionofanysinglycircumferentia lclosedplanecur eroundanya isinitsplanenotcuttingit. A

renchusage isaringgeneratedby there o lutionofacircle

notcuttingit. Anysimplering oranysolidwith

maybecalledatoroid buttodeser ethisappella tionit

nli eatore .

ofatoroid isaline throughitssubstancepassingsomewhatappro imatelythroughthecentresofgra ityofallits crosssections.An

fatoroidis anyclosedlinein itssurfaceonceroundits

tionofa toroidisanysectionby aplaneorcur edsurface

intotwoseparatetoroids.Itmustcut thesurfaceof

impleclosedcur es oneof themcomplete ly surroundingthe

ce:of courseitisthe spacebetweenthesecur eswhich

toroidalsubstance andtheareaofthe inneroneof

perture.

cuttinge eryaperturalcircumference eachonceand

ss sectionofthetoroid.It consistsessentiallyofasimple



s s_ u s e # p d



rcularcy linder. Letabetheradiusof

thea iso f theclosedheli ; le trdenote

tionoftheideal toroidonthesurface

supposedsmallincomparisonw itha and

noftheheli tothenormalsectionof

w ere thestepof thescrew and2 rris

lindricalcore onwhichanyshortpart

telysupposedtobewound.

stant Ithegi enforceresultanto f the

g e n r ot a ti o na l m om e nt . W e h a e ( ~ 2 8

I N 7V r T 2 ( a .

r=

eadofasingle threadwoundspirally

w eha etw oseparatethreadsforming asit

edscrew " andleteachthreadma eaw hole

oroidal core.Thetwothreads each

ically tortuousringslin edtogether and

w hatw illnow beadoublev orte - ring.

btainedfora singlethreadwouldbe

ifK cdenotedthecyclicconstantforthe

ds ortwicethecyclic constantfor

rof turnsofeitheraloneroundthe toroidal

enienttota eNforthenumberof

othatthenumberof turnsofonethread

thecyclicconstantfore itherthreada lone

steadymodesof thedoublev orte - ring

c N 7r r2 a

odeswillcorrespondto smallerand

butinthiscase asinthecaseof thesingle



s s_ u s e # p d



adsruntogetherintoone asil lustratedforthe

ne e d d ia g ra m ( f i g 7 .

F i g . 8. " T r ef o il K n o t.

oftheli uidroundtherotational

suchthatthef luid- e locityatany

dinthesamedirectionas theresultant

spondingpointin theneighbourhood

cuit o rga l aniccircuits o f thesameshape

etting-forthofthisanalogytopeople

tura listsare w iththedistribution o f

bourhoodofan electriccircuit does

nderstandingofthestillsomewhat

whichweareatpresent occupied.

onofthe li uidinther otational

eceof Indian- rubbergas-pipestif fened

usualmanner andwithitconstruct

chw eha ebeen

thesymmetricaltre fo ilk not( f ig. 8 ~ 1 ) , unit ingthetwo

y by ty ingthem

twoofstra ightcy lindrica lplug thenturnthetuberound

nuousa is. The

uidvorte -core

titmustbe

d K n o t .

rform ofthe

cularto theplaneofthediagram

isthroughthecentreof thediagram

ane ineachofthecasesrepresented

s.Thewholemotionofthe fluid

issorelatedin itsdifferentpartsto



s s_ u s e # p d



andsmallin diameterperpendicularto

e itmaybe. Icannot how e er sayat

atthispossiblesteadymotionis stable

f re o lution de iatinginf inite ly litt le

hthesamevorticity there isthesame

rgy andconsiderationofthegeneral

notreassuringonthepoint ofstability

tioniswanting .

deedsucceededin rigorously

yoftheHelmholt ringinanycase.

imaginetheringtobethic erinoneplace

hegi env orticity insteadofbe ing

alcircularring tobedistributedin a

is butthinnerinonepartthaninthe

ththesamev orticityandthesame

hsucha distributionisgreaterthanwhen

ut nowletthefigureof thecross

teadofbe ingappro imatelycircular be

.Thiswilldiminishthe energywiththe

meimpulse.Thusfrom thefigureof

ntinuouslytootherswithsamev orticity

meenergy. Thus w eseethatthef igure

tedabo e a f igureofma imum-minimum

imum norofabso luteminimumenergy .

mum-minimumproblemw ecannotderi e

enaofsteam-ringsandsmo e-rings

tw ere thenatura lhistoryof thesub ect

andthatthesteadyconf iguration w ith

metersofcoreto diameterofaperture

ationsconnectedwithwhatisrigorously

ostabil ityofv orte co lumns( tobegi en

the RoyalSociety mayleadtoa

stabilityforasimpleHelmholt ring

roportiontodiameterofaperture.B ut

istorynor mathematicsgi esusperfect

nthecrosssection isconsiderablein

perture.

W . T . M ay 1 0 1 88 7.



s s_ u s e # p d



statementofgeneralpropositions

plesusedintheprecedingabstract o fw hich

sof papersonv orte motioncommunicatedtotheRoyalSocietyofEdinburghin 1867 -68and-69

actionsfor 1869.Therestwillform

continuationof thatpaper w hichIhope

alSocietybeforethe endofthepresent

ha i n g v o r t e m ot i on i s c al l ed v o r te c o re o r f o r br e i t y s i mp l y " c o r e. A n y fi n it e p or t io n o f li u i d

re andhascontiguousw ithito eritsw ho le

ingli uid isca lledav orte . A v orte

aring ofmatter.Thatitmustbe so

dpublishedbyHelmholt . Sometimesthe

ndedto include irrotationallymo ingli uid

ingintheneighbourhoodofv orte -core

of li u idmaysuccessi e lycomeintothe

e andpassaw ayagain w hile thecore

yofthesamesubstance itismoreproper

termavorte asinthedef init ionIha e

ationofav orte isthecirculation[ V . M. ~ 60( a ] inanyendlesscircuitoncerounditscore .

n fi g ur a ti o ns a v o r te m a yt a e w h et h er o n

diness(~ 1abo e , oronaccountof

ort ices orbyso lidsimmersedinthe

d bo u nd a ry o f t he l i u i d ( i f t h e l i u i d is n o t

ua ti o n r e ma i ns u n ch a ng e d [ V . M . ~ 5 9 P r op . ( 1 ] .

e issometimescalleditscyclic constant.

nethroughaf luidmo ingrotationally

r ed w hosedirectionate erypo int

fmolecularrotationthroughthatpoint

] .

o r t e i s e ss e nt i al l y a cl o se d c ur e [ b e in g

v orte ] * .

edsectionofavorte isany

normally thea ia ll inesthroughe ery

d e le t ed b y L or d K e l i n .



s s_ u s e # p d



losedsectionofavorte intosmaller

sthroughthebordersof theseareasform

-tubes. Isha llca ll( a f terHelmholt ) a

ortionofav orte boundedbyav orte tube( notnecessarily inf initesimal . O fcourseacompletev orte

orte - f i lament butit isgenera lly

stermonly toaparto favorte asj ust

yofacompletev orte satisfiesthe

tube.

be isessentia llyendless. Inav orte f i lamentinf inite ly smallina lldiametersofcrosssections" rotation

0 ( e f r om p oi n t to p o in t o f th e l en g th o f t he

etotime in erselyastheareaof thecross

area ofthecrosssectioninto the

circulationorcyclicconstantof the

to designateinageneralwaythe

otationinthemattero fav orte .

orte di idedintoanumberof infinitely

thev orticityw illbecomplete lygi enw hen

mentanditscirculation orcyclicconstant

hapesandposit ionsof thef ilamentsmust

erthatnotonly thev orticity butitsdistribution canberegardedasgi en.

yatanypo into favorte isthecirculationofaninf initesimalf i lamentthroughthispo int di ided

ompletefi lament. Thevorte -density

dforthesameportionoffluid.B y

lla longanyonev orte - f i lament.

into infinitesimalf i lamentsin erse lyas

the ircirculationsaree ua l andletthe

ofunity . Ta ethepro ectionofa ll

.1/nof thesumofthe areasofthese

M . ~ 6 6 2 e u a l to t h e co m po n en t i mp u ls e

iculartothatplane. Ta ethepro ections

anes atrightanglestoone another

ityof theareasof thesethreesetsof

accordingtoPoinsot smethod theresultant

oupleof thethreeforcese ua lrespecti e ly to



s s_ u s e # p d



as andactinginlinesthroughthethree

ndiculartothethree planes.Thiswillbe

eforceresultanto f the impulse andthe

orte .

stosay thecouple isalsocalledthe

v o r te ( V . M . ~ 6 .

mentofaplane arearoundany

eareamultipliedintothedistancef rom

dicularto itsplanethroughitscentreof

f thepro ectionofaclosedcur eon

eaofpro ectionisama imumwillbe

ctionof thecur e orsimply theareaof

hepro ectiononanyplaneperpendicular

ntarea iso fcoursez ero .

tanta iso faclosedcur e isa line

ity andperpendiculartotheplaneof

tantarealmomentof aclosedcur e

resultanta iso f theareasof itspro ectionsontw oplanesatrightanglestooneanother andpara lle l

tood o fcourse thattheareasof the

oplanesarenote anescentgenera lly

aplanecur e andthatthe irz ero- a lues

fe ua lposit i eandnegati eportions.

tin generalz ero.

edefinitions theresultantimpulse

of inf inite ly smallcrosssectionandofunit

heresultantareaof itscur e . The

rte isthesameastheresultanta iso f the

ona lmomentise ua ltotheresultantareal

ntav orte - f i lamentinaninf inite

ginfluenceofothervortices oro f so lids

Wenow see f romtheconstancyof the

era lly inV . M. ~ 19 , thattheresultantarea

mentofthe cur eformedbythe

tnthowe eritscur emaybecomecontorted anditsresultanta isremainsthesameline inspace.



s s_ u s e # p d



YNA MIC S [ 10

tionsandcontortionsthevorte - f i lamentmay

anymotionof translationthroughspacethis

eragealongtheresultanta is.

ua lv orte madeupofaninf inite

orte - f i laments. If thesebeofv o lumes

othe irvorte -densit ies(~ 25 , sothatthe ir

w enow seef romtheconstancyof the

eresultantareasofall thev orte filamentsremainsconstant andsodoesthe sumoftheirrotational

ntareala isofthemall regardedasone

nspace. Hence asinthecaseofav orte f i lament thetranslation if any throughspace isonthea erage

A llthis o fcourse isonthesupposit ion

rte , andnoso lidimmersedinthe li uid

of the li uid nearenoughtoproduceany

gi env orte .



s s_ u s e # p d



N A L M O T I O N O F A L I Q U I D

AT S .

o l . x v . 1 8 77 p p . 29 7 -8 C om m un i ca t ed t o S ec t io n A

onatGlasgow September7 1876.

gthismotionwerelaidbeforethe

la ined buttheanaly tica ltreatmentof

more mathematicalpapertobecommunicatedtotheSectionon Saturday.Thechiefob ectofthe

astoillustratee perimentallyaconclusionfromthistheorywhichhad beenannouncedbytheauthor

theSection , to theef fectthat if the

an oblatespheroidalrigidshellfullof

multipleoftherotationalperiodof the

ofthe spheroidisofthe difference

eastdiameters theprecessionaleffect

ontheshe ll isappro imately thesame

drotatingwiththe samerotational

mentconsistedinshow inga li uidgy rostat

of thinsheetcopperfilledwithwater

dfly-wheelofthe ordinarygyrostat.

e hibited thee uatoria ldiameterof

dedthepo lara isbyaboutone- tenthof

peedtobethirty turnspersecond

w hich if actingonarotatingso lidof the

s w ouldproduceaprecessionha ingits

t ipleof Io fasecond must accordingto

eryappro imately thesameprecessionin

uidasinthe rotatingsolid.Accordinglythemainprecessionalphenomenaoftheli uidgyrostat

entfromthoseofordinarysolid gyrostats

A ddresses( Macmillan , v o l. II. pp. 2 8-272.

9



s s_ u s e # p d



onforthesa eofcomparison. It is

r ationwithoutmeasurementmight

rencesbetweentheperformancesofthe

statinthewayofnutationaltremors

caseof theinstrumentwiththefist.

nteitherofspeedsor forceswas

tion andtheauthormerelyshowedthe

ghgenerali l lustration w hichhehoped

nterestingil lustration o f thatvery

ematicalhydro- inetics thequasi- rigidity

i uidby rotation.

tionofthispapertothe Association

peningaddresswhichprecededit onthe

e i edf romProf . HenryNo. 240of theSmithsonianC ontributionstoKnow ledge o fdateOctober 1871 entitled

MotionpresentedbytheGyroscope the

o e s a nd t h e Pe n du l um b y B r e e t M a o r G e n. J . G . B a r n ar d C o l. o f E ng i ne e rs U . S . A . i n w hi c h I fi n d a

nofmypre iously-publishedstatements

ccasionofmyaddresstocorrect e pressed

donotconcur withSirW illiamThomsonintheopinions

f romThomsonandTa it ande pressedinhis

ope( Nature F eb. 1 1872 . Sofaras

erfectrigidity withinaninfinitelyrigid

in therateofprecessionw ouldbeaf fected.

perGen. B arnardspea sof " the

byrotation. Thushehasanticipated

mentscontainedin mypaperonthe

farasregards theeffectofinteriorfluidity

nofa perfectlyrigidellipsoidalshell

tererroro f thatpaper w hichI

dress hadnotbeencorrectedby

attheplausiblereasoningw hichhadledme

mcon incing. F ormyself Icanonly

yearliestopportunity tocorrecttheerrors

erors andthatIdeeply regretany

doneinthe meantime.



s s_ u s e # p d



S

uidGyrostats.Thesolidgyrostat

ormanyyearsin theNaturalPhilosophy

tofGlasgow asamechanicali l lustrationof

o lids andithasa lsobeene hibitedin

on ersa ionesof theR oya lSocieties

graphEngineers butnoaccountofit

efollowingbriefdescriptionand

ennowbeacceptableto readersof

essentiallyofa massi efly-wheel

to f inertia pi o tedonthetw oendsof its

dtoanoutercasew hichcomplete ly inclosesit. F ig. 1representsasectionbyaplanethroughthea is

ig. 2asectionbyaplaneatrightanglesto

roughthecase j ustabo ethef ly -w heel.

d withathinpro ectingedgeinthe

whichiscalledthe bearingedge.Its

cur il inearpo lygonofsi teensidesw ith

hefly-wheel.Eachsideof thepolygon

radiusgreaterthanthedistanceof the

efriction ofthefly-wheelwould if

cular causethecasetoroll alongonit

opre entthisef fectthatthecur edpo lygonal



s s_ u s e # p d



ndrepresentedinthedrawingisgi ento

apieceofstoutc ordaboutfortyfeet

clearrunofaboutsi ty feetcanbeobta inedarecon enient. Thegyrostatha ingbeenplacedwiththe

ertica l thecordispassedinthroughan

o-and-a-halftimesroundthebobbin-shaped

utagainatan apertureontheopposite

rethattheslac cordisplacedclearofa ll



s s_ u s e # p d



S

eef romk in s theoperatorholdsthe

case ispre entedf romturning w hile

throughbyrunning atagradually

y f romtheinstrument w hileho ldingtheend

ufficienttensionisappliedtothe

titf romslippingroundontheshaf t. In aa s

- - -- - --

ngularv e locity iscommunicatedtothef lyw heel suf ficient indeed tok eepitspinningforupw ardsof tw enty

en spunitbeset onitsbearing

a itye actlyo erthebearingpo int on

nesuchasa pieceofplate-glasslyingona

nueapparently stationaryandinstablee uil ibrium. Ifwhile it isinthisposit ionacoupleroundahori onta l

f ly -w heelbeappliedtothecase no

mthevertica lisproduced butitro tates

a is. If ahea yblow w iththef istbe

case it ismetbyw hatseemstothesenses

stiff e lasticbody and fora few seconds

statisinastateofv io lenttremor w hich

rapidly . A stherotationa lve locitygradua lly

ofthetremorsproducedby ablowalso



s s_ u s e # p d



ioustonoticethe totteringcondition

asied tremulousnessof thegy rostat w hen

asedtospin.

fly-wheelisreplacedby anoblate

sheetcopper andf il ledw ithwater. The

heinstrumente hibitedisI- thatisto

iametere ceedsthepo larby thatf ractionof

thetw oendsof itspo lara isinbearings

brasssurroundingthespheroid.This

nnectedwiththecur edpolygonalbearingedgewhichliesinthe e uatorialplaneoftheinstrument

forthesupporto f thea iso f the

asectionisrepresentedthroughthe

ty andF ig. 4gi esav iew of thegyrostat

hepro longationof thea is. Topre ent

nthegyrostatfallsdownatthe endofits

tdroundit insuchaw ay thatnoplanecan

eli uidgyrostatissimilarto that

ostat dif feringonly intheuseofavery

largewheelforthe purposeofpulling

onabobbin f reetorotaterounda

sthenpassedtwo-and-a-halftimes

nin theanne edsectionaldrawing

ecircumferenceofthe largewheelto

stantthenturns thewheelwithgradually

whiletheframeofthegyrostatis firmlyheld

onappliedtotheenteringcordtopre ent

pulley.



s s_ u s e # p d



5 ]

ETS[ ILLUSTR A TINGV O R TEX -SYSTEMS .

l . x v I II . 18 78 p p. 1 , 1 4.

ericanJ ournalofScience describing

ngmagnetsbyMrA lf redM. Mayer to

mofmutually-repellentmoleculeseach

owardsafi edcentre whichappearedin

. p. 487 mustha e interestedmanyreaders.

ularlybecausethemodeofe perimentingtheredescribed w ithaslightmodification gi esaperfect

easilyreali edwithsatisfactoryenough

thek inetice uil ibriumofgroupsofco lumnar

ncirclesroundthe ircommoncentreofgra ity

ctofacommunicationI hadmadetothe

honthepre iousMonday.InMr

ori onta lresultantrepulsionbetw eenany

saccordingtoacomplicatedfunctionof

lycalculableifthe distributionof

wereaccuratelyk nown.Supposethe

ysimilarin allthebarsand ineachto

nglaw:-Letthe intensityofmagnetisationberigorouslyuniformthroughoutav erylargeportion

of thebar( F ig. 1 , andletitvaryuniformly

ndsA andB . Thebarw illactasif

bstitutedidealmagneticmatter , or

a lled uniformlydistributedthroughthe

thew holequantity inDB tobee ua l

nk indtothatofC A . F ore ample

arityinAB andtruesoutherninB D.

neednotbee ual.LetnowA C D B '

lectrostaticsandMagnetism ~ 469( W. Thomson .



s s_ u s e # p d



actlysimilardistributionofmagnetism

dletthetw obeheldpara lle ltooneanother.

aryin erselyasthedistance ifthe

ll incomparisonw ithDB orC A andif

mall incomparisonwithCD.Ifthe

werfulbar-magnetbeheldina linemidwaybetweenB A andB ' A , a tadistancef romtheendsB andB '

sonw ithB B ' andcomparablew iththe

thehori onta lcomponentof itseffecton

cev aryingdirectlyasitsdistance from

rthesecondit ionsMrMayer se periments

fe uil ibriumof two orthree orfour

ints inaplane repellingoneanother

themutua ldistances andeachindependentlyattractedtowardsaf i edcentrew ithaforcev ary ingdirectly

sIshowedin mycommunicationtothe

h istheconfigurationofthegroup of

eofstraightcolumnarv orticeswith

byaplaneperpendiculartothe

ertiaof agroupofidealparticles of

esepointsbe ingthef i edcentre inthe

tyreferredtoby MrMayerhas

numerica lproblem andit isremar able

y orinstabilityisidenticalin the

ms.Inthestatic problemitisofcourse

fthe mutualforcesbetweentheparticles

attractiontowardsaf i edcentre isless

blee uilibriumthanforanyconfigurationdifferinginfinitelylittlefrom it.Thepotentialenergy

a functionofdistancefromthecentral

tance increases andthestatementof

enientlymodif iedtothefo llow ing: oragi env a lueof thisfunctionthemutua lpotentialenergy

minimumforstablee uil ibrium. When

heattracti e forcevariesdirectlyasthe

nergyis

onstants andZ r2thesumof thes uaresof



s s_ u s e # p d



MAGNETSILLUSTR A TINGV O R TEX -SYSTEMS 1 7

rticlesfromtheattracti ecentre.And

eenthe particlesisthein ersedistance

ergy ise ua lto

...

ot e co ns ta nt s a nd D D , D , & a m p c . d en ot e th e mu tu al

ticles.Thus

uilibrium becomesthattheproductofthe mutual

ticlesmustbe

i e n v a l u e of t h e

e irdistancesf rom

f irstconclusion

ethat the

particlesmust

eNow thecondit ionofk inetice uil ibriumofagroup B a

thatistosay the

o l e incircles n

eof inertia is D

unicationtothe

h thatthe

ancesmust

mumorama imum-minimumforagi env a lueof the

e irdistances

ofgra ity ; and

netice uil ibriummaybestable isthattheproduct

env a lueof

so f the irdistances

Ta ingfor

tices( oro f the little C

Mayer sproblem , s '

uil ibrium is -

ented by

hecaserepresented g. 1.

headsinF igs. 2and3 representthemotions

dthatw hate erbethecomplicationofmotionsduetomutual

rt ices the ircentreofgra itymustremainatrest. [ O f .

s up ra .



s s_ u s e # p d



sroundthe ircentreofgra ity . Itmustbe

feachco lumnre o l esa lsoroundits

hesamedirectionasthegrouproundthe

tyofa ll w ithenormouslygreaterangular

dtheproblemofoscillationsin the

^ â € ” '

g. 3 .

rationofstablee uilibrium.Thegeneral

tsformathematicalanalysishasa v ery

orthecaseofatriadofe ua lvorte

oodoftheanglesof ane uilateral

git k inematicallyisrepresentedin

circulardiscsofcardboardpi otedonpins

anglesofane uilateraltriangle

ne.Theplanecarryingthesethree

entlymadeofacirculardisc ofstiffcard



s s_ u s e # p d



N A L O S C I L LA T IO N S O R O T A T IN G W A T ER .

gsof theR oyalSocietyofEdinburgh March17 1879:

o l . x , 1 8 80 p p . A0 ~ l / ]

sub ectinhisDynamicalTheoryof

dea ltw ithinitsutmostgenera litye cept

themotionofeachparticleto be

tal andtheve locity tobea lw ayse ua l

ev ertical.Thisimpliesthatthe

allin comparisonwiththedistance

dtof indthede iationf romle e lnessof the

sensiblefractionofits ma imum

ortcommunicationI adoptthis

ther insteadofsupposingthew atertoco er

fthesurfaceof asolidspheroidasdoes

mplerproblemofanareaofw atersosmall

ureof itssurface isnotsensiblycur ed.

anyshape andofdepthnotnecessarily

atest smallincomparisonw iththe least

dthe waterinitrotateround a

ularve locityw sosmallthatthegreatest

itmaybe asmallfractionofthe radian:

gularv e locitymustbesmallincomparison

rgdenotesgra ity andA thegreatestdiameter

ionsofmotionare

,

componentve locit iesofanypo into f the

mnthroughthepo int( x y , re la ti e ly

, O y r e o l i n g wi t h th e b as i n p t h e p r es s ur e

o f thisco lumn andptheuniformdensity



s s_ u s e # p d



onsw emustha ey= a/ V -1 w herea isrea l.

ations.

thetessera ltypewemustha e

n V / - 1 w he r e m an d n a re r e al a n d by p u tt i ng

ginaryconstituentswefind

.. . .. . . .. . . .. . 10 ,

onnectedby thee uation

. . .. . . 11 .

ngva luesofuandv , w eseew hatthe

tbetoallowthesetesseraloscillations

shape. Nobounding- linecanbedrawn

hehori ontalcomponentve locityperpendicularto it iszero . Thereforetoproduceorpermitoscil la t ions

ein respecttoform watermustbe

ternatelyallroundtheboundary or

a ll f o rw hichthehori onta lcomponent

ero.Hencetheoscillationsofwater

ughare notofthesimpleharmonic

andtheproblemoffindingthemremains

thew ell- now nsolutionforw a esina

hichareofthe simpleharmonictype.

t ionsinanendlessC analw ithstraight

para lleltox , thesolutionis

a t . . .. . .. . .. . .. . .. . . ( 1 2 ;

t h e e u a t i on

1 ) ,

1 1 .

w ef indthatv v anishesthroughoutifwe

14 ;



s s_ u s e # p d



A T IO N A L O S C I L LA T IO N S O R O T A T IN G W A T ER 1 4 5

1 i n ( 1 2 w e fi n d b y ( 7 ,

- a t . . .. . .. . .. . .. . . 1 5 ;

1 ) w e fi nd

theve locityofpropagationofwa esis

riodasinaf i edcanal. Thusthe

inedto theeffectofthefactor

tingresultsfollowfromtheinterpretation

particular suppositionsastothe

odof theoscil la t ion(27r/ & lt r , theperiod

) a n d th e t im e r e u i re d t o tr a e l a t th e

anal.Themore appro imatelynodal

henorthcoastofthe EnglishChannel

nchcoast andof thetidesonthewest

nnelthanon theeastorEnglish

accountedforontheprinciplerepresented

intoaccounta longw ithf rict iona lresistance

es oftheEnglishChannelmaybe

orepow erfulw a estra e ll ingfromw est

esspowerfulwa estra e ll ingf romeastto

outhernpart oftheIrishChannelby more

ll ingf romsouthtonorthcombinedw ithless

ll ingf romnorthtosouth. Theproblemof

ndlessrotatingcanal issol edbythe

H e - ly c o s ( m - a t - e y ^ ( c o s m + a t }

- at + E l yc os ( m + a - t } . .. 1 7 .

cana l w efa llupontheunso l edproblem

eraloscillations.Ifinstead ofbeing

posethecanalto becircularandendless pro idedthebreadthofthe canalbesmallincomparisonwith

theso lution( 17 sti l lholds. Inthis

cumferenceof thecana l w emustha e

aninteger.



s s_ u s e # p d



W A V E S I N CI R CU L A R B A S I N ( P O L A R

( i - a t . . . .. . . .. . . .. . . .. . . .. 18

w hereP isafunctionof r. B y ( 8 ) P

on

. .. .. .. . 1 9 ;

d

r/ P

P. .. .. . 2 0 .

2cr- r

erinacircular basin withor

and.Let abetheradius ofthe

centralislandleta be itsradius. The

fulf i lledare = 0w henr= aand

onetotheotherof thetwoconstants

andthespeed* aof theoscilla t ion are

ntit iestobefoundby thesetwoe uations.

isimmediatelyeliminated andthe

e uationfora.Thereis nodifficulty

nthusf indingasmanyasw epleaseof the

ndw or ingoutthew holemotionof the

tsof thise uation w hicharefoundto

er-Sturm-Liou il le theory arethespeeds

mentalmodes correspondingtothedifferent

onsof the idiametra ldi isionsimpliedby

Thus bygi ingto ithesuccessi ev a lues

a m p c . a n d so l i n g th e t ra n sc e nd e nt a l e u a ti o n so f o un d

efundamenta lmodesofv ibrationof the

posedcircumstances.

rthreetida lreportso f theB rit ishA ssociationthew ord

ncetoasimpleharmonicfunction hasbeenusedtodesignate

abodymo inginacircle inthesameperiod. Thus if T

thespeed v iceversa if abethespeed 27r/ ristheperiod.



s s_ u s e # p d



A T IO N A L O S C I L LA T IO N S O R O T A T IN G W A T ER 1 4 7

d theso lutionof ( 19 w hichmust

chPandits differentialcoefficientsare

nceP isw hatisca lledaB esse l sfunction

orderi and accordingtotheestablished

. ( 2 1 .

.. . .. . 21

eforanendlesscircular canalisfallen

ygreatv a luetor. Thus ifw eput27rr/ i= X

wa e- length w eha ei/ r= 27/ X , w hichw ill

otation.We mustnowneglecttheterm

ndthusthedif ferentia le uationbecomes

0

.............................. 22

or2-4w 2 / gD. A so lutionof thise uationis

a - r a n d us i ng t h is i n ( 2 0 a b o e w e fi n d

i n ( m - a t ( a l - 2o m e - l y w h er e m = i O .

a t e a c h bo u nd a ry w e h a e a l = 2 w m w h ic h

lyattheboundaries butthroughoutthespace

atee uation( 22 issuf f icientlynearly true.

v a l u e ab o e w e ha e

g- - o) , )

a b o e .

nicationtotheRoyalSocietytogo

ses andtogi edeta ilso f theso lutionsat

eofwhichpresentgreatinterest inrelation

sinre lationtotheabstracttheoryofvorte

differencesbetweencasesinwhichar

than2oareremar ably interesting and

ectto thetheoryofdiurnaltidesin

d e r B e s se l s c he n F u n c ti o ne n ( L e i p i g 1 8 67 , ~ 5 a n d

d ie B e s s e l s c he n F u n c t io n en ( L e ip i g 1 8 68 , ~ 2 9 .



s s_ u s e # p d



YNA MIC S [ 1

thermoreorless nearlyclosedseasin

thelunarfortnightlytide ofthewhole

edthattheprecedingtheoryis applicable

sinanynarrow la eorportionof thesea

afew degreesof theearth ssurface if f o r

entof theearth sangularv e locity rounda

ality -thatistosay w = ysinI w herey

gularv e locity andIthe latitude.



s s_ u s e # p d



itherinouridea lin iscidincompressible

uchasw aterora ir to formavorte -sheet

aceof f initeslip byanynatura laction.

at presentunderconsideration andin

blecaseof twoportionsof li u idmeeting

instance adropof ra infa ll ingdirectlyor

ntalsurfaceofsti l lw ater , isthatcontinuity

uid motionbecomeestablishedatthe

eentwopoints orbetweentwolinesin

mmetrytowhichourpresent sub ect

theseparationof the li uidf romthe

osedglobeorany otherfigureperfectly

is andmo inge actly inthe lineof the

esof thef reedli uidsurfacecomeintocontact

heenclosureoftworings ofv acuum

ich how e er maybeenormously farf rom

on .

ne-integraloftangentialcomponentv elocityroundanyendlesscur eencirclingthering asaring ona



s s_ u s e # p d



C O R E L E SS V O R T IC E S

slin edtogether isdeterminateforeach

andremainsconstantfore eraf ter: unless

ormore orthetw of irstformedunite into

dentsthereis nosecurity.

e thatacore lessring- orte , w ith

nditshollow shallbeleftoscillating in

e u a to r o f th e g lo b e p r o i d ed ( V 2 - P / P

terialoftheglobebe v iscouslyelastic

steadyposit ionroundthee uator ina

calonthetwosidesof thee uatorial

motiongoesonsteadilyhenceforthfore er.

c e ed a c e rt a in l i mi t I s u pp o se c o re l es s

si e ly formedandshedof fbehindtheglobe

uid.

possibil itybe impossible foraglobe it iscerta inly

prolate figuresofre olution.



s s_ u s e # p d



A CO L U M N AR V O R T EX .

gsof theR oyalSocietyofEdinburgh March1 1880

pp. 155-168.

ion inwhichthestream-linesare

w iththe ircentresinoneline( thea iso f

eve locit iesappro imate lyconstant andappro imatelye ua late ua ldistancesfromthea is. A sapre lim inary

enienttoe pressthee uationsofmotion

pressiblein iscidfluid( thedescriptionof

t in estigationisconf ined intermsof

s " r 0 z - t ha t i s c o or d in a te s s uc h t ha t

y .

y andifw edenotebyx , y z the

fthefluidparticlewhichattime tispassing

y z ) , a n d by d / dt d / d , d / dy d / d d i f f er e nt i at i on s r es p ec t i e l y on t h e su p po s it i on o f x , y z c o ns t an t t y z

c o ns t an t a n d t x , y c o ns t an t t h e or d in a ry e u a ti o ns

~ 2

I

2 .

y..................... 2



s s_ u s e # p d



A C O L U M NA R V O R T EX

arcoordinates weha e

3 )

onsare

d r d r

d ( r O ) d r O )

= . . . . . .. . . .. . . . (5 .

pro imate ly incirclesroundO z , w ith

appro imatelye ua ltoT a functionof r

s assume

t - i0 , r 0 = T + r c os m c os ( n t -i

t - i , p = P + w co sm c os ( n t - iO )

dw arefunctionsof r eachinfinitely smallin

tutingin( 4 and( 5 andneglecting

f the infinitely smallq uantit ies w ef ind

T

( 7 ,

w

...............



s s_ u s e # p d



inatinga andresol ingforp T wefind

T dT

+ - d- w

w iT 2 f dT 2 _. T 2 2

-i -- -+ n ---w

dr r d r r

- + - n-

o fm= 0 ormotionintw odimensions

enienttoput. .. . . .. . . .. . . .. . . .. . . .. . . .. 10 .

hichsuperimposedon = 0andrO = T

tionisirro tational andfbsin( nt- i is

sa lsotoberemar edthat w henm

superimposedmotionisirrotationa lw here if

= c o n s t. / r a n d th a t wh e ne e r i t is i r ro t at i on a l b c o s m s i n ( n t - i , w i th b a s g i e n b y ( 1 0 , i s i ts

8 by ( 9 , w eha ea lineardif ferentia le uationof thesecondorderforw . The integrationof this

u lt i n ( 9 , g i e w p a n d r in t e rm s o f

yconstantsof integrationw hich w ithm n

minedtofulf i lw hate ersurface-condit ions

conditionsofmaintenanceareprescribed

nterestingcasespresentthemsel es.

esttobegin:CASEI.

o const. . . . .. . . .. . . .. . . .. . . .. 11 ,

i 8 w h en a p pr o i m at e ly r = a

t i , , , , r = ai . 1

a b ei ng a ny g i e n q u an ti ti es a nd i



s s_ u s e # p d



initelysmallsimpleharmonicnormal

es distributedo erthemaccordingto

nrespecttothecoordinatesz, 0. The

shadby supposingtheinnerboundary

, andthe li uidcontinuousthroughoutthe

ercylindric boundaryofradiusa.

ma esw= O whenr= 0 e ceptforthe

lly w ithoute ception re uiresthatcbe

orw becomes

. .. . .. . . . . . .. . .. . .. . .. ( 1 9 ,

i( or . . . .. . . .. . . .. . . .. . . .. . . . 20 ;

w he n r = a g i e s b y ( 1 ) ,

1 ,

n- iwe a

mula.

mannerofF ourier w ef indtheso lution

ionof thegenerati edisturbanceo er

o ereachof thetw oifw edonotconf ine

se andforanyarbitraryperiodicfunctionof

r edthat(6 representsanundulation

nderw ithlinearv e locityna/ ia tthe

elocityn/ithroughout.Tofindtheinterior

rationproducedatthesurface w emust

, o ranysumofso lutionsof thesametype

so lutions ina llrespectsthesame e cept

dthatgreatenoughv a luesof ima e

hereforev imaginary andforsuchtheso lutions

) ifunctionsmustbeused.

a lV orte inaf i ed

a

ed o r bi t r = a + f r a d t. .. . 2 2 ,

sturbedorbit r= a+ f rdt



s s_ u s e # p d



) o f th is s ur fa ce w eh a e f ro mn P = T 2 d r/ r o f ( 6

) . .. .. . ( 3 2 .

a n d ( 2 6 , a n d ( 2 5 , a n d ( 2 ) , t h e co nd it io n

ndarygi es

o ,

i ( m a ] + ( c ( m a + [ 01 ( ma ] = 0 ......... 3 3 ) .

by ( 27 , w egetane uationto

wef ind

.. . .. . .. . .. . .. . .. . .. 3 4 ,

lyposit i enumeric.

a se i s t ha t o f a= o o w hi c h b y ( 2 7 ,

th er ef or e b y( 3 3 ) , g i e s

. .. . . .. . . . 3 5

Subcase w eseethatthedisturbance

a ellingroundthecylinderwithangular

o ( 1 - V N / i ,

superimposedononeanother tra e ll ing

ularv elocitiesgreaterthanand

thantheangularve locityof themassof the

sbye ua ldif ferences. Thepropagation

e locity isinthesamedirectionasthatin

es thepropagationof theotherisinthe

&gt i2( asitcertainly isinsomecases .

edinmotionwithoneor otherof

e locit ies( 3 4 , orlinearv e locities

a n d th e l i u i d be t h en l e ft t o i ts e lf i t w il l p er f or m

uatorymo ementrepresentedby ( 6 ,

. B u t i f t h e fr e e su r fa c e be d i sp l ac e d to t h e co r ru g at e d



s s_ u s e # p d




henlef tf reee itheratrestorw ithanyother

elocitythaneitherofthose thecorrugation

ntotw osetsofw a estra e ll ingw iththe

aco(1 + IN/ i .

ceptional andcanpresentno

undthecylinder. Itwillbe considered

rlyimportantandinteresting.To

ma r t h at

.. . .. . .. . .. . .. . . 6 .

n mr = I o ( mnr

f ( 24 is

( m 2r 2 2 4 4+ & a m p

2 + & a mp c

ntsandSi= 1-+ 2-1+ . . . i-1. Hence

” 1 - + & a mp c .. .. .. .. .. .. . 3 7 ,

t a e n s o as t o m a e 1 o ( 0 1 .

edthere lationbetweenEandDtoma e

andfoundittobe

= + 2 - 0 79 4 42 - 1 9 6 5 1 0 = 1 15 9

( 3 8 .

41244881...........................

nterna lF rict ionofF luidsontheMotionofPendulums "

n d ( 1 0 6 . ( C am b . Ph i l. T r an s . D e c . 1 85 0 .

. W. L. Gla isherthatGauss insection3 2ofhis

r al e s ci r c aS e ri e m In f in i ta m 1 + ( a . P / l . y x + & a m p c . ( O p e r a

e s th e v a lu e of - r r 2 o r - ( - p , i n hi s no ta ti on t o

6 51002602142 47944099.

stfigureinSto es sresult( 106 ought asinthete t

lle t sTablesw ef ind

98 592825

rnumberf romthis w eha ethev a lueofE/Dto20places



s s_ u s e # p d



nientassumptionforconstantfactor

+ 2 + 4 + & a m p c .

4 r 4

+ ' ( S 2 + 11 59 ) + & am p c .

t t he s e ri e s in ( 3 6 a n d ( 3 9 a r e

ergreatbemr thoughforv a luesofmr

micon ergente pressions w il lgi ethe

earlyenoughfor mostpracticalpurposes

llabour.

3 9 w e fi n d b y d i ff e re n ti a ti o n

5

4 + 2 4. 6 + & a mp c .. .. .. . ( 4 0 .

4 r4 & a m p c

2- + 2 2. .+ 2 + & a m p c .4 6

- 1+ / ( S + 1 15 9 1 5 ]

2 1 15 9 1 5 ] + & a mp c .

5r |

4 6 + & a m p c .

[ - 3 + 1 .2 ( S + ' 1 15 9 1 5 ]

S 2+ 1 15 9 1 5 ] + & a mp c .

2 2 42 6 + & a m p c .

eSubcasew ithi= 1 supposemato

i n g th a t S = 1 w e h a e

1 + 2 - lo gm a- + ' 1 15 9 ( m a 2 .. .. .. 4 2 .

-a + - 1 15 9. .. .. . 4 2 .



s s_ u s e # p d




rofv a luesofX y ie ldedby (50 gi es

a n d ( 1 ) , a s o l u ti o n of t h e pr o bl e m of f i nd i ng s i mp l e

aco lumnarv orte w ithmofanyassumed

eharmonicv ibrationsarethusfound:

hemannerofF ourier fo rdif ferentv a lues

plitudesanddif ferentepochs gi ese ery

atinginfinitelylittlefromtheundisturbed

thatofi= 0 iscuriouslyinteresting.

1 , ( 52 gi e

m a : , , â € ” ) m al o( m ) . .. .. .. .. .. .. .. .. . 5 ) ,

E ( m a

. . . .. . . .. . . .. . . .. . . .. . . ( 54 .

5 ) , regardedasatranscendentale uation

1 st 2 nd 3 r d . .. r oo ts o f J ( q ) = 0 i n or de r of

a n d th e n e t g r ea t er r o ot s o f J , ' ( q ) = 0

dow ntotherootsof J thegreaterthey

tedbyaid ofHansen sTablesof

o a nd J 1 ( w h i ch i s e u a l to - J o f ro m q = 0

a isasmallf ractionofunity thesecond

a largenumber ande enthesmallestroot

fr a ct i on t h e fi r st r o ot o f J o ( q ) = 0 w h ic h

Table is2 4049 or appro imate lyenough

I n e e r y ca s e in w h ic h q i s v e r y l a rg e i n

hethermaissmallornot ( 54 gi es

6 , w eseethatthesummationof two

espropagatedalongthelengthofthe

n t- m ) r O = T + T C 7o s 7 1t - -M ) )

; p = P+ i c os ( n t - m ) (

tionofthesewa esisn/m.Hence when

nw ithma theve locityof longitudina l

/ o f thetranslationa lve locityof thesurface

rbit. Thisis1/ 1 2 or5of thetrans R epublishedinL6mmel sB esse l scheF unctionen Le ip ig 1868.



s s_ u s e # p d



hecaseofmasmall andthemodecorrespondingtothesmallestrooto f ( 5 ) . A fulle aminationof the

e a s e p r e ss ed b y( 5 5 , ( 1 ) , ( 4 8 , ( 1 5

tructi e . Itmustformamorede eloped

alSociety.

andmav erysmall isparticularly

tInitw eha e by ( 42 , fo rthesecond

p pr o i m at e ly

og I + 11 59 ] . .. 5 6 .

m a

oot q , iscomparablewithma anda ll

mparisonwithma.Tofindthesmallest

sv erysmallw eha e toasecondappro imation

4 1 . .. . .. . .. . .. . .. . .. . ( 5 7 .

1 b e co m es t o a f ir s t ap p ro i m at i on

. .. .. 5 8

dtofindthetwo un nownsX andq 2 gi e

m 2 a2

. No w w i th i = 1 ( 5 1 b e co me s

elysmall. Hence( 52 gi esfora

. . . . . .. . . .. . . .. . 59

2. . .. . . .. . . .. . . .. . . 60 .

- ~ . . .. . .. . .. . .. . .. V .

5 9 , ( 6 0 , a nd ( 5 6 i n( 5 0 , w ef in d t oa se co nd

4 3 m 2a ( 3 )

g mc + - 1 1 59 ,

- m2 2a g + 4 3 * 2 ma

a log + + -1159. . .. . . .. . . .. 61 .

s 6 1 .



s s_ u s e # p d




4 ) a b o e . T h e f ac t t ha t a s i n ( 4 ) ,

, showsthatinthiscasea lsothedirection

a elsroundthecylinderis retrograde

hetranslationoffluidin theundisturbed

a s wa s t o be e p e ct e d t h e v a l ue s o f - n ar e a pp r o i m at e ly e u a l in t h e tw o c as e s wh e n ma i s s ma l l en o ug h b u t - n

y sm a ll d i ff e re n ce i n ( 4 t h an i n ( 6 1 a s

.

gt 1hasaparticularlysimple

rthesmallestq -rootofthetranscendental

u e o f i in s te a d of u n it y w e st i ll h a e ( 5 8 ,

forq small. E lim inatingq2/ m2a2betw een

t i ll f i nd X = ; b u t i n st e ad o f n = 0 by ( 5 1 , w e

w . Th u s is p r o e d t he s o lu t io n f or w a e s o f

f iguretra e ll ingroundacy lindrica lv orte ,

agowithoutproofinmyfirst article

tion .

s " P r o c. R o y. S o c. E d in . F e b . 18 1 8 67 [ s u pr a p . t I 6 .



s s_ u s e # p d



S T EA D Y AN D O F P E RI O D I C F L U I D M O T I O N 1 6 7

ppositionwill helpinanin estigation

hichwillfollow.

completelyenclosedinac ontaining

eeitherrigidorplasticsothatw emayat

hape orofnaturalsolidmaterialand

tic(thatistosay returninga lwaystothe

entimeisallowed butresistingalldeformationswithaforce dependingonthespeedofthe change

eofq uasi-perfectelasticity .Thewhole

elandfluidwill sometimesbeconsideredas

disturbedbygra ityorotherforce

pposeitto beheldabsolutelyfi ed.

emaysuppose ittobeheldbyso lidsupports

iscouslye lastic materia l sothatitw il l

esenseasathree-leggedtableresting

Thefundamenta lphilosophicquestion

paramountimportance inourpresentsub ect.

e pla inedinThomsonandTait sNatural

n PartI. ~ 249 andmorefullydiscussed

oninapaper" On theLaw of Inertia the

andthePrincipleofAbsoluteClinural

ion. F orourpresentpurposeweshall

sumingourplatform theearthorthe

absolute ly f i edinspace.

hepresentcommunication so farasit

istopro eandto il lustratetheproof

ositionsregardingamass offluidgi en

arto f it: I Theenergyof thewholemotionmaybeinf inite ly

inacertainsystematicmanneron the

ngingit ultimatelytorest.

esse lbesimplycontinuousandbe

sticmateria l thef luidgi enmo ingw ithin

- esse lbecomple lycontinuousand

elasticmateria l thef luidw ill loseenergy

er buttoadeterminatecondit ionof irro tationa l

substantia lly inP roc. B oy . Soc. Edin. xnII. 1885 p. 114

r gy i n V o r t e M o ti o n. ]



s s_ u s e # p d



atecyclicconstantfor eachcircuit

r e ma r , f i rs t t h at m e re d i st o rt i on o f t he

shapeof theboundary canincreasethe

e ly. F orsimplicity supposeaf initeoran

hapeofthe containing- esseltobe

me thiswilldistorttheinternal

ha edoneif thef luidhadbeengi en

H el m ho l t ' s l a ws o f v o r t e m o ti o n w e c an

ialstateofmotionsupposedk now n the

erypartof thef luid a f terthechange.

eshapeof thecontaining- esse lbealteredby

hatistosay dila teduniformly inone or

ns andcontracteduniformly intheother

ofthreeatright anglestooneanother.

eneouslydeformedthroughout thea is

chpart willchangeindirectionso as

ngdirectionofthe samelineoffluid

tudewillchangein in ersesimpleproportiontothedistancebetweentwoparticles inthelineof the

mplif y subse uentoperationstotheutmost

yq uic motionorbyslow motion the

ngedtoa circularcylinderwithperforated

s asshowninfig.1. Inthepresent

of the li uidmaybesupposedtoha e

tyofmolecularrotationthroughout.It

omomentofmomentumroundthea iso f

allsupposethisnottobethecase. If it

w hichcouldbedisco eredbye ternal

ecy linder roundadiameter toaf resh

eitwithmomentof momentumofthe

iso f thecy linder. Withoutfurther

w esha llsupposethecy lindertobegi en w ith

containingf luidinane ceedingly irregular

thagi enmomentofmomentumMfround

Thecy linderitself istobeheldabso lute ly f i ed andthereforew hate erw edotothepistonswe

mentofmomentumofthefluid round



s s_ u s e # p d



S T EA D Y AN D O F P E RI O D I C F L U I D M O T I O N 1 6 9

epistonA tobetemporarily f i edinits

dthewholecontaining- esselofcylinderandpistons to

sspi ot soas

A A thea iso f

elbeofideally

itsinnersurfacebe

o l ut i on i t w il l

remainatrest

efluid onitis

ththea is. B ut

ea lly rigid le tthe

scous-e lasticso lid

ssoftheinternalfluidmotionwill cause

ning-solidwithlossofenergy andthe

tedtomoreandmorenearlyas time

heonedeterminateconditionof minimum

momentofmomentum w hich asisw ell

ed isthecondit ionofsolidandf luid

larve locity . If thestif fnessof the

allenoughandits v iscositygreatenough

alconditionwill becloselyappro imatedtoinav erymoderatenumberoftimestheperiodof

n.Still wemustwaitaninfinite

perfectappro imationtothiscondition

mple orirregularinitialmotion.W e

cuttheaf fa irshortbysimplysupposingthe

gw ithuniformangularv e locity l i ea

g- esse l a truef igureof re o lution w hich

ras absolutelyrigid andconsistingof

iaphragmandtw omo ablepistons as

ullo rpushandlea e itto itse lf it

ce inthedirectionof the impulseand

eepalternate lypullingandpushingit

isstatementrecei esaninterestinge perimentali l lustration

e tractedfromtheProceedingsoftheRoyalInstitution

ch4 1881 be inganabstracto faF riday-e eningdiscourse

aspossiblyaModeofMotion andnow inthepressfor

therlecturesandaddressesinav o lume( V o l. r. o f the



s s_ u s e # p d



gularoscillation duetosuperposition

accordingtoaninfinitenumberof

thek indin estigatedinmyarticle

mnarV orte , " P roc. R oy . Soc. Edin.

not asthere l im itedtobe inginf initesimal

bev iscouslyresistedthesev ibrations

wn andiftimeenoughisallowed

beenimpartedtothe li uidby the

sw illbe lost anditw illaga inberotating

asitw asinthebeginning.

U M E N E RG Y I N V O R T E M O T I O N .

ymotionof anincompressible

ite fi edportionofspace( thatistosay

ocityanddirectionofmotioncontinue

ntofthespace withinwhichthefluidis

gi env orticity theenergy isathorough

ughminimum oraminima . Thefurther

red bytheconsiderationofenergy

eadymotionfor whichtheenergyisa

thoroughminimum becausew henthe

heenergy iso fnecessityconstant. B ut

energydoesnotdecide theq uestion

steadymotioninwhichthe energyisa

mmencingw ithanygi enmotion

edindefinitelybyproperly-designed

( understoodthattheprimiti eboundary

w ithgi env orticity butw ithnoother

horoughma imumofenergy inanycase.

ptinthecase ofirrotationalcirculationina

e s s el r e fe r re d t o in ~ 3 ( I I I a b o e b e

eenergybyoperationonthe boundary

it i eboundary , asw eseeby thefo llowing

acommunicationreadbeforetheB rit ishA ssociation SectionA atthe

urday A ugust28 1880 andpublishedintheR eportforthat

nNature O ct. 28 1880. R eprintednow withcorrections

ions.

o 9 a bo e .



s s_ u s e # p d



N D MI N IM U M E N E RG Y I N V O R TE X M O T I O N 1 7

p a ra l le l a n d op p os i te l y ro t at i ng v o r te

endicularlybytwofi edparallelplanes.

thecylindricboundary theymay in

otion bethoroughlyande uablymi ed

.In thisconditiontheenergyis

lt ring reducedbydiminutionof its

gtube coiledwithintheenclosure.

yisinfinitelysmall.

co lumn w ithtwoendsontheboundary

ynearlymeetstheboundary and

dedtil l it isbro enintotwoe ualand

mns connected oneendofonetooneend

nishingv orte l igamentinfinitelynearthe

herdealtwithtill thesetwocolumnsare

ua lannihila tion.

present thee tremelydif f icultgenera l

orsuggested by theconsiderationofsuch

esnow totwo-dimensiona lmotionsina

edparallelplanesanda closedcylindric

dric surfaceperpendiculartothem

f f igure(buta lw aystrulycy lindricand

es . A lso forsimplicity conf ineourse l esforthepresenttov orticitye itherpositi eorz ero ine ery

iousthat withthelimitationtotwodimensionalmotion theenergycannotbeeitherinfinitelysmall

ygi env orticityandgi ency lindric

egi encondit ions therecerta inlyare

motions-thoseofabsolutema imum

ergy.Theconfigurationofabsolute

yconsistso f leastvorticity ( orz ero

f luidofz erovorticity ne ttheboundary

orticityinwards.Theconfigurationof

yclearlyconsistsofgreatestv orticity

ndlessandlessvorticity inwards. If there

rticity a llsuchf luidw illbeatreste ither

orinisolatedportionssurroundedby

. F oril lustration seef igs. 4and5

eninsosimpleacaseasthatof the



s s_ u s e # p d



sentedinthediagram therecanbean

teadymotions eachw ithma imum

a imum energy anda lsoaninf inite

otionsofminimum( thoughnotleast

ninfinitenumberofconfigurations

hof themha ingtheenergyathorough

12 , w esee byconsideringthecase in

ryofthe containing-canisterconsists

unicatingbyanarrowpassage as

ch acanisterbecompletelyfilled

gf luidofuniformv orticity thestreamlinesmustbesomethingli e those indicatedinf ig. 4.

tportionof thew holef luidisirro tational it isclearthattheremaybeaminimumenergy and

ationofmotion withthewholeof

tso f thecanister orthew hole in

proportioninoneandtherestintheother.

m-lines

owninf ig. 4.

n asshown

he



s s_ u s e # p d



N D MI N IM U M E N E RG Y I N V O R TE X M O T I O N 1 7 5

rationofma imum energy forwhich

i uidisintw oe ualparts inthe

f theenclosure.Thereisaninfinite

ofma imumenergyinwhichthe

isune uallydistributedbetweenthe

osure.

o w hentheboundary iscircular f f^ '

ntriccirclesand thefluidisdistri-

riclayersofe ua lvorticity . In the

umenergy thev orticity isgreatestat

andislessandlessoutw ardstothe

emotionofminimumenergythe

hea is andgreaterandgreateroutw ards

presstheconditionssymbolically

thef luidatdistancerf romthea is

ectionofthe motionisperpendicularto

andletabetheradiusof theboundary . The

pressiondiminishesf rom r = 0tor= a

ndofma imumenergy . If it increases

themotionisstableandofminimumenergy.

hes ordim inishesandincreases asr

themotionisunstable .

e le tthevorticitybeuniform

of thew holef luid andz erothrough

emotionof greatestenergy the

orticityw illbe intheshapeofa

i easo lidrounditsow na is

nearlyreachedintheyear1875 byrigidmathematical

brationsofappro imatelycircularcy lindricvortices but

blicationofitbyLord Rayleigh whoconcludeshis

ty orInstabil ity o fcertainF luidMotions ( P roceedingsof

Society F eb. 12 1880 w iththefo llowingstatement: Itmaybepro edthat if thef luidmo ebetw eentw origidconcentricw alls the

idedthatinthe steadymotiontherotationeithercontinually

ecreasesinpassingoutw ardsf romthea is " -w hichw as

me(A ugust28 1880 w henImadethecommunicationto

AssociationatSwansea.



s s_ u s e # p d




dofinertia. Thewholemomentof

a 2 - 2 b 2 - M

fthisamountas longastheboundary

nsumptionofenergystillgoeson

oesonisthis: thew a esofshorter

tipliedande altedtilltheircrests run

uid andthoseofgreaterlengthare

rtionof the irrotationally re o l ing

iththecentralv orte column.The

atmaybeca lledav orte spongeis

omogeneous ona largescale butconsisting

dirrotationalfluid moreandmore

stimead ances. Themi ture isa ltogether

reof thew hiteandyellow ofanegg

ell- nownculinaryoperation.Letb

dricv orte sponge andg itsmean

chisthesamein allsensiblylargeparts.

1887. - Iha ehadsomedif f iculty innow pro ingthese

nd18 o f1880. Here isproof . Denotingforbre ity1/ 27rof the

u and1/ 27rof theenergybye w eha e

T r. r dr a n d e= | T f 2 .r d r.

eleastpossible sub ecttotheconditions:( 1 thatAthasa

t ha t

0

r= a T=Pb2/ a thislastconditionbeingtheresultanto f

hetota lvorticity ise ua ltothatof ' uniformw ithinthe

sdescribedinthe lastthreesentencesof~ 17andthe

yso l etheproblemw hen

2- b2 ; o rt -& g t - a2 .

18sol esitwhen

o r = b o a 2.

18sol esitwhen

2- b2 ; or & l t ' b 2a2 .



s s_ u s e # p d



the radiusoftheoriginalv orte column

0tor= b ,

2 /r f ro m r= b t o r = a

b/ b

2 + -

thecylindriccasef romgoingroundin

dingituntil somemoremomentof

mthefluid.Thenlea eitto itself

ongew illsw ellby theminglingw ithito f

tationalli uid. Continuethis

cupiesthe wholeenclosure.

cessfurther andtheresultwill be

ngcanister isallowedtogoround freely

endtoacondit ioninw hichacerta in

rte coregetsf i lteredintoaposit ionne t

dadistancef romthea isw hichwesha ll

thef luidw ithinthisspacetendstoamoreand

tureofvorte w ithirrotationa lf luid. This

onrepetit ionof theprocessofpre enting

ound andagainlea ingitf reetogo

andmorenearly irrotationa lf luid andthe

becomesthic erandthic er. The

ow

r & l t c

t c

ntumis

( a 2 - C2 } .

swhichthewholetends isabeltconstitutedof theoriginalv orte corenowne ttheboundary and

e o l edirrotationa lly rounditnow

eingthecondit ion( ~ 16abo e o f

y.B eginoncemorewiththecondition

a bs o lu t e ma i m um e ne r gy a n d le a e t h e fl u id



s s_ u s e # p d



N D MI N IM U M E N E RG Y I N V O R T EX M O T I O N 1 7 9

hecanisterfreeto goroundsometimes

pro idedonly it isult imate lyhe ldf rom

theultimateconditionisalwaysthesame

16 o fabsoluteminimumenergy. Theenclosingrotationa lbe lt be ingtheactualsubstanceof theoriginal

nitssectionalareatorrb2 andthereforec2=a2-b2.

m isnow -7r~ b4 beinge ualtothe

theportionoftheoriginal configuration

ntra lvorte .

e eninimagination thevery f ine

nanddrawing-outoftherotational

ingwiththe irrotationa lf luid andits

mtheirrotationalfluid whichthe

18hasforcedonourconsideration. This

andw esubstituteforthe" v orte sponge

somerespectsmore interesting conception

q uitearbitrarily andmere ly tohe lpusto

energy-transformationofv orte column

eattributetotherotationa lportionof the

alattractionbetweenitsparts" insensible

andbetweenitandtheplaneendsof the

suchrelati eamountsastocausethe interfacebetweenrotationalandirrotationalfluidtomeettheend

ettheamountofthisLaplacian

ly small- sosmall f o re ample thatthe

tchthesurfaceof theprimiti ev orte

timesitsareais smallincomparison

enf luidmotion. E ery thingw illgo

17 18if insteadof " runoutinto f ine

1 7 l i ne 2 9 w e s ub s ti t ut e " b r e a o f f in t o

orte co lumns ; andinsteadof " sponge

ute" spindrif t .

mmenergy forgi env orticity

momentum(thoughclearlynotuni ue

becausemagnitudesandordersof

nsofconstituentcolumnsof the

shitf romthe" New tonian" attraction because Ibe lie e

oroughlyformulated" attractioninsensibleatsensible

dedonita perfectmathematicaltheoryofcapillaryattraction.



s s_ u s e # p d



varied isfullydeterminateastothe

o lumnre lati e ly totheothers andthe

esasif itsconstituentco lumnsw ere

scouslye lasticcontainingvesse l each

describedin~ ~ 17 18 f l iesroundw ith

cityasthespindrif tcloudw ithin andso

nstably w ithoutlossofenergy untilthe

nstoppedor otherwisetamperedwith.

hattheLaplacianattractionwould

e co lumnstobrea intodetacheddrops( as

w ncaseofaf inecircularj eto fw ater

nwardsfromacirculartube andwould

fw atergi enatrestinaregionundisturbedbygra ity butitcouldnot becausetheenergyof the

efluidround thev orte columnmust

mncould brea inanyplace.The

how e er ma ethecy lindricform

cludedf romallsuchconsiderationsat

12 totw o-dimensiona lmotion.

nattractionand returntoour

fincompressiblefluidactedononly

gsurface andbymutualpressure

byno" appliedforce" throughitsinterior.

to fmomentumbetw eenthee tremepossible

2 a n d 7r t b4 t h er e i s cl e ar l y b e si d es t h e 1 7 1 8

egy anotherdeterminatecircularso lution

nofcircularmotionofwhichtheenergy is

er circularmotionofsamev orticity

mentum.Thissolutionclearlyisfound

intotwoparts-oneacircularcentra l

o theracircularcy lindricshell l in ingtheconta iningvesse l theratioofoneparttotheotherbe ingdetermined

otalmomentofmomentumha ethe

hisso lution( assa idabo e ~ 14andfootnote maybepro edtobeunstable.

e amongotherillustrationsof

sub ectdemandingseriousconsiderationandin estigation notonlybypurelyscientificcoercion

acticalimportance.



s s_ u s e # p d




oncludew iththecompleteso lution

e solution( onlyfoundwithinthe

r~ 10-18of thepresentarticlew ere

blemonwhichI firstcommencedtrials

nergyanabsolutema imum intw odimensionalmotionw ithgi enmomentofmomentumandgi en

anistero fgi enshape. Theso lutionis

uni ue " absolutema imum meaning

ms. B utthesamein estigationincludes

roblem: Tof ind o f thesetsofso lutions

ferentconf igurationsof themotionha ing

entum.F oreachofthesethe energy

otthegreatestma imum forthegi enmoment

nterestingfeatureofthepractical

enowattainedisthe continuoustransitionfromanyonesteadyorperiodic solution throughaseriesof

s toanyothersteadyorperiodicsolution

eof operationeasilyunderstood and

rol.Theoperatinginstrumentismerely

o lumn orrod f ittedperpendicularlybetween

andmo ableatpleasuretoanyposit ionpara llel

re . It isshow n mar edS inf igs.

entingthesolutionofourproblemforthecase

asmallpartof itswholev olume

uid tow hiche igencyof timelimitsthe

orte lininguniformlytheenclosing

n thecentreofthestill waterwithin

ocityof thew aterinthevorte increases

toSb2/ aattheoutside incontactw iththe

thenotationof~~ 15and16. Now mo e

omitscentralposition andcarryitround

elocity& lt ~ b/ aand& gt Ib/ a . A dimple

beproduced runningroundalitt le in

butult imately fa ll ingbac tobemoreand

thestirreris carrieduniformly.If

yslowedtillthedimplegetsagain in

andisthencarriedroundinasim ilar

ra lw aysa litt lebehindtheradiusthroughthe

heangularve locityof thedimplewill



s s_ u s e # p d



sdepthanditsconca ecur aturewill

theangularv e locity isr b/ a thedimple

atis theenclosingwall w ithitsconca ity

inf ig. 7 andtheangularv e locityofpropagationbecomes~ ' b/ a .

essvorte be ltnow becomesdi ided

thetwoac uiredendsbecomerounded

at er .

i g . 7.

arrowheadsreferstothev elocityofthestirrerand of

e locityof thef luid.

evorte re fertove locityof f luid. A rrow headsinthe irrotationa lf luidrefertothestirreranddimple . A rrow headsinabcreferto

relati elytothedimple.

F i g . 9.

ertomotionof thestirrer andof thevorte asaw hole.

ottedcirclerefertoorbita lmotionofc thecentreof the

nfullf inecur esrefertoabso lutev e locityof f luid.



s s_ u s e # p d



NTA LILLUSTR A TIONO F MINIMUMENER GY18

arriedroundalwaysalittlerearward

ofabreastthemiddleofthe gap.F igs.

continuingthe processtillultimately

entra landcircular( w ithonly the infinitesimal

senceof thestirrer withwhichweneed

tpresent .

tanystageof theprocess a f tertheformationof thegap thestirrertobecarriedforwardtoastation

ofabreastthemiddleof thegap or

erearof thev orte ( insteadofsomew hatinad anceof thefrontasshow ninf ig. 8 . Theve locity

gmented( by rearwardpull ) , the

llbediminished:thev orte trainwill

eachesroundtoits rear eachbeing

andbroughtintoabsolutecontactwiththe

drear uniteinadimplegradually

processmaybe continuedtillweendas

rte l in ingthe insideof thew alluniformly

emiddle ofthecentralstill-water.

LILLUSTR A TIONO F MINIMU MENER GY.

I I i. N o . 1 8 1 8 80 p p . 69 - 70 B r i t i sh A s so c ia t io n R ep o rt

g. 3 0 p p . 49 1 -2 .

a li uidgyrostatofe actlythe

describedandrepresentedby the

printed from Nature F ebruary1 1877

ththedif ferencethatthef igureof theshell is

hee perimentwasinfactconducted

w hichw ase hibitedtotheB rit ish

876 alteredbythesubstitutionof a

toria ldiameterabout-9of itsa ia ldiameter

iameter29ofe uatoria ldiameterw hich

tuswasshownas asuccessfulgyrostat.

lswereeach ofthemmadefromthe

copperwhichplumberssoldertogether

f loaters. B ya litt lehammeringit iseasy

the propershapestoma eeitherthe

.



s s_ u s e # p d



tthe rotationofali uidina rigid

e ingaconf igurationofma imumenergy for

uldbeunstable if theconta iningv esse lislef t

rfectlyelasticsupports althoughit

- - - - -- -

esse lwereheldabso lute ly f i ed orborne

ts orlefttoitself inspaceunactedon

ditw asto illustratethistheory thatthe

df il ledw ithw aterandplacedinthe

efirst trialwasliterallystartling

a ebeenso asitw asmere lya

nanticipatedbytheory.The framewor washeldasfirmlyas possiblebyonepersonwithhis two



s s_ u s e # p d



NTA LILLUSTR A TIONO F MINIMUMENER GY185

steadyashecould. Thespinningby

passingroundasmallV pulleyof i- inch

theo a lshe llandroundalargef ly -w heel

dattherate ofaboutoneroundper

edforse era lm inutes. Thisinthecaseof

sk now nf rompre iouse periments w ould

f f icientrotationtotheconta inedwaterto

ctw ithgreatf irmnessli easo lid

erimentw iththeo a lshe ll theshell

thgreatv elocityduringthelastminute

momentitwas releasedfromthecord

f ramew or inmyhands Icommenced

ontalglasstabletotest itsgyrostatic

r w hichIhe ldinmyhandsga eav io lent

dinafewseconds theshellstopped

thepi otshadbecomebento er by

intheneighbourhoodofthestiff

so lderedto it show ingthatthe li uidhad

coupleaga institsconta iningshe ll ina

theef forttoresistw hichbymyhands

shellwasrefittedwith morestrongly

hee perimenthasbeenrepeatedse era l

ecideduneasinessof thef ramew or is

nho ldingit inhishandsduringthe

asthecordiscutandthepersonho lding

perimenta ltable thef ramew or

ow riggleroundinhishands andby thetime

onthe tabletherotationisnearly all

gyrostatispreciselywhatwas e pectedfromthetheory andpresentsatrulywonderfulcontrast

w iththeapparatusandoperationsine ery

ptinha inganoblate insteadofaprolate

d.

ongcordf irstw oundonabobbin andf inallyw oundup

helargew heelasdescribedinNature F ebruary1 1877

founditmuchmorecon enienttouseanendlesscordlitt le

rcumferenceof thelargewheel andlessthanhalfround

pulleyof thegy rostat andtok eepitt ightenoughto

ntialforceontheV pulley isdesiredby thepersonho ldingthe

ftercontinuingthe spinningbyturningthefly-wheelfor

gedproper theendlesscordiscutw ithapa iro f scissors



s s_ u s e # p d



G I N I N IT Y I N LO R D R A Y LE I GH S S O L U T I O N

P L AN E V O R TE X S T RA T U M .

I I I. 1 8 80 p p . 45 - 46 B r i t i s h As s oc i at i on R e po r t S wa n se a

1 88 0 p p. 49 2- .

E IG H S s o lu t io n i n o l e s a f or m ul a e u i a l en t t o

ma imumvalueof they -componentof

enotesaconstantsuchthat27r/ m isthew a elength , Tdenotesthetranslationa lve locityof thev orte stratumw henundisturbed w hichisinthex direction andisafunctionofy , ndenotesthev ibrationalspeed oraconstantsuch

isstable if ononeside it isbounded

dif thevorticity (orv a lueof IdT /dy

ideally f romthisplane e ceptinplaces

nstant.

supposeafi edboundingplaneto

rpendiculartoOy andletdT/ dyha eits

0 anddecreasecontinuously orbyone

f romthisva lue toz eroaty=aand

.

t he w a e - e l oc i ty w h at e e r b e th e

mediatebetweenthegreatestandleast



s s_ u s e # p d



O R T E X S T RA T U M

certa inva lueofybetw een0anda

y ise ua ltothew a e- e locity or

alueofythe secondtermwithinthe

sformula isinf initeunless forthe

T/ dy2v anishes.

considerationof thisinf inity ifw eta e

constantv orticity ( dT/ dy=constant

asforthiscasetheformula issimply

einfinity whichoccursinthe more

suggestsane aminationof thestreamlines byw hichitsinterpretationbecomesob ious andw hich

hecaseofconstantv orticity themotionhas

cterat theplacewherethetranslational

ew a e- e locity . Thispeculiarity isrepresentedby theanne eddiagram w hichismosteasilyunderstood

tona lve locit iesaty= 0andy= ato

andofsuchmagnitudethatthew a e elocity isz ero sothatw eha ethecaseofstandingw a es.

- l inesareasrepresentedintheanne ed

gionof translationalv elocitygreater

alvelocity isseparatedfromtheregion

lessthanw a e-propagationalv e locityby

ernofellipticwhirls.



s s_ u s e # p d



andTait sNaturalPhilosophy that

nto fmomentum there isone andonlyone

uil ibrium.

minthere o lutiona lf igure isstable or

f = - - - ) i s & l t o r & g t 1 - 9 4 57 .

mentofmomentumislessthanthatw hich

oreccentricity= -81266 forthere o lutiona l

tonly stable butuni ue.

mentofmomentumisgreaterthanthat

9457forthere o lutiona lf igure there is

o lutiona lf igure the J acobianf igurew ith

w hichisa lw aysstable if thecondit ionofbe ing

ut asw illbeseenin( f be low the

houttheconstra inttoell ipsoida lf igure isin

able thoughitseemsprobablethat in

houtanyconstraint.

msonandTa it sNaturalPhilosophy ~ 778

aagreatmultipleofb w eseeob iously

o ecmustinthiscasebev erysmallin

weha eav eryslendere llipso id longin

appro imate lyapro latef igureof re o lution

a i s w h ic h r e o l i n g wi t h pr o pe r a ng u la r

esta isc isa f igureofe uil ibrium. The

w hich w ithoutanyconstra int is inv irtue

tionofminimumenergyorofma imum

momentofmomentum isaconf iguration

enmomentofmomentum sub ectto

eis constrainedlyanellipsoid.F rom

iseasilyverif ied inthe lighto f~ 778of

uralPhilosophy itfollowsthat withthe

hee uil ibriumisstable . There o lutiona l

w iththesamemomentofmomentum

( 1 , ( 2 , a n d ( 3 ) , w i ll b e f ou n d in t h e fo r th c om i ng n e w

ait sNaturalPhilosophy V o l. I. PartII. [ Theproofsw ere

neralproblemhasbeenanaly edinacomprehensi e

fLordK el in smethods byH. Po incard inaclassica l

n I . 18 8 5. C f . La mb s H y dr o dy n a i c s C h . x n I .



s s_ u s e # p d



F E Q U I L I B R IU M O F R O T AT IN G F L U I D

eroid forittheenergy isaminima ,

mallestenergythatare olutional

omentofmomentumcanha e butit

oftheJ acobianfigurewiththesame

fbeinge ll ipso idalisremo edandthe

it isclearthattheslenderJ acobian

stable becauseade iationf romellipso ida l

ngit inthemiddleandthic eningit

f thic eningit inthemiddleandthinningit

ouldw iththesamemomentofmomentumgi e

h sogreatamomentofmomentumasto

slenderJ acobiane ll ipso id it isclearthat

e uil ibriumistw odetachedappro imately spherica lmasses rotating( asifpartsofaso lid roundan

eof inertia andthatthisf igure isstable .

aybe aninfinitenumberofsuchstable

proportionsof the li uidinthetwo

hesamemomentofmomentumthere

e uil ibriumw iththe li uidindi ers

wodetachedappro imatelyspherical

oninmorethantwodetachedmasses

dingtothedef init ionof ( Ic be low

ranyof them e enifundisturbedby

ouldha etruek ineticstability ata lle ents

caseof thethreematerialpoints

heau( seeR outh sR igidDynamics ~ 475

mthestablek inetice uil ibriumofa

ua lorune ua lportions so farasunder

tely spherica l butdisturbedtoslightly

by thew ell- now nin estigationofe uil ibriumtides gi eninThomsonandTait sNaturalPhilosophy ~ 804 ,

prolatefigureswhichwouldresultfrom

outchangeofmomentofmomentum

latef igures now note enappro imately

stable ispeculiarly interesting. Weha e

weentheunstableJ acobianellipsoid



s s_ u s e # p d



YNA MIC S [ 19

ty andthecaseofsmallestmoment

withstabilityintwoe ualdetached

nofhowtofill upthisgapwith

mostattracti eq uestion tow ards

ntoffer nocontribution.

ergyw ithgi enmomentofmomentumis

imum thek inetice uil ibriumis

uidisperfectly in iscid. Itseems

allyunstablewhentheenergyisa

k now thatthisproposit ionhasbeene er

i s c o si t y h o w e e r s li g ht i n t he l i u i d o r

ye lasticso lid how e ersmall f loating

thee uil ibriuminanycaseofenergy

a imumcannotbesecularly stable

econfigurationsarethose inwhich

withgi enmomentofmomentum.

a inwhetherw ithgi enmomentof

morethanonesecularlystable configuration

ousf luid inonecontinuousmass butit

thereis onlyone.



s s_ u s e # p d



9 )

A L I Q U I D W I T H IN A N E LL I PS O I D A L

gsof theR oyalSocietyofEdinburgh V o l. x III.

- 7 8.

cedthepropositionsregardingfluid

lhollowwhichformthe sub ectofthe

andw hich thoughob iousenoughand

donotseemtoha ebeenpre iouslydisco ered.

nhomogeneousrotation orhomogeneous

esignatetheconditionofa fluidinrespectto

outittheamountsof itsmolecularrotation

iallinesparallel.Thisdesignationclearly

tingsolid:butitis applicableofcourse

seofaf luid inw hichirrotationalmotion

mogeneousrotationasofasolid.To

otionthussignified considerthefollowing

h i ch ( 1 a n d ( 2 a r e in c lu d ed i n ( 3 ) : 1 L e t a li u i d k e p t i n th e s ha p e of a f i gu r e of r e o l ut i on

sse l begi eninastateofhomogeneous

of thef igure . Letanimpulsi erotation

rtothisa isbegi entotheconta ining

usmotionof the li uid atthe instant

eted consistsofanirrotationalmotion

enhomogeneousrotationalmotion.The

uiddoesnotgenerallyremainhomo



s s_ u s e # p d



T H I N A NE L LI P SO I D A L H O L L O W

heresultantof thisirrotationalmotion

enrotationalmotion.Theirrotational

phericalhollowisof courseeasily

nownsphericalharmonicanalysisforfluid

eonlytheinstantaneousmotion which

n theimpulseiscompleted.Theinfinitely

w or ingouttheconse uencesaccording

ns asto force orastochangingshape

notfollowat present.Itwillbefully

nw hichtheboundaryof the li uidis

egin with andisconstrainedtobe

a l. Itwil lbepro edthatinthiscasethe

uidremainsalwayshomogeneous.

thegeometrica l" stra in isessentia lly

tali uidcontainedwithinachanging

pro idedthatthemotionof thef luidbe

orbeatanyoneinstanthomogeneously

ousnessofthegeometricalstrainbeing

w sf romHelmholl ' sfundamentalprinciplesof

atthemolecularrotationmustcontinuehomogeneous itsmagnitude w henthere isanystretchingorcontraction

v ary ingin erse lyasthe lengthofa lineof

ction andthea ia ldirectionvary ingso

thesamesubstantialline.

ia tionf rome actnessinthee ll ipso ida lf igure thehomogeneousnessof therotationof the li uidis

here isno lim ittotheamountofde iation

whichmaysuper eneinconse uenceof

entotheboundary w hetherinthe

orofmotionwithoutchangeofshape.

thepresentto motionoftheboundary

w ef indit interestingtoremar thatw e

easingorindefinitelydiminishingthe

by properlyarrangedactionintheway

ngv esse l. Tocontinua lly increasethe

low ingrulemaybecorrect a lthoughIdo

fofit.Supposethecontainingv essel

ndthe li uidw ithinittoha eperfectly

thinthenote actlyellipsoidalhollow

maybegintomo eoritmaynot.



s s_ u s e # p d



semi-a es. F rom( 4 w ef indforthe

2-a2 a-b2

a ~ Y

C 2. 6

a 2

utionof theproblem sofarasconcerns

cityatanypo into f thefluid w hichis

noughinthesolutionof ahydrodynamical

e interestingine erycase andit iseasy

eitupto thedeterminationoftheposition

uidatany time andw emaytherefore

l y to t h e a e s o f t h e el l ip s oi d l et ( x p i

e t o fanyparticularparticle$ of the

v e l oc i ti e s ( d / d t d p /d t d / d t o f t he

y tothee ll ipsoidaree ua ltothedif ferences

s ( u v , w , o f t he a b so l ut e v e l o c it y o f $ ,

mponentsoftheabsolutev elocityofan

z ) rigidlyconnectedw iththeell ipsoid and

~) atthetimet. These lastcomponentsare

p X . . .. .. . .. . .. . .. 7 .

, a t t h e i ns t an t ( x , I ) c o i n c id e wi t h

eha e

8

2 ~ + a 2 ' a

ale uationsofthefirstorder fordetermining( x , i i intermsoft. Denotingd/dtby8 wemaywrite



s s_ u s e # p d



I T H I N AN E L LI P SO I D A L H O L L O W 1 9 9

o . . .. . . .. . . .. . . .. . . .. . 9

ntheusua lmannerw ef ind - / = 0. . .. . . .. . . .. . . . 10 ;

hedeterminantandremo ingthesuperf luousfactor8 w eha e

.....

a bc ( x + 7 +

;

thirdof ( 9 w eha e

X . . . .. . .. . .. . .. . ( 1 4 .

may ta eastheso lutionforanyoneof the

ample asfo llowsX = A coscot

b2 ) + c - ( 2 a 2 + ( . 2 I J



s s_ u s e # p d



YNA MIC S [ 20

e s

b 22

.

e plicitlythepositionofany chosenparticle

seitwould beeasytofindfromthem what

siertodothisf romtheunintegrated

tiply ingthef irsto f thesebya/ a2 thesecondby 3 / b2 andthethirdbyy / c2 andadding w ef ind

1

rbit l iesintheplane

.. . . .. . 18 ,

t.

o fe uations( 8 byx / a2 thesecond

by / / c2andadding w ef ind

1 9

a e

. .. . . .. . . .. . . .. 20 ,

ant.

tli esontheellipsoid( 20 ; andwe

heellipseinwhichthis ellipsoidiscut

e plicitfully integratedso lution( 15

hataparticleof thef luiddescribes re lati e ly

hichthef luidisconta ined thee ll ipse

20 , accordingtothe law ofasingleparticle

the influenceofaforcetowardsa fi ed

proportiontodistancefrom thecentre.



s s_ u s e # p d



A NDSMA LLOSC ILLA TION O F A PER EC T

N E AR L Y ST R AI G HT C O R E L ES S V O R TI C ES .

rtoPro fessorG. F . F it Gerald. F romtheProceedings

my readNo ember3 0 1889.

rmedonethingIw asgo ingtowritetoyou

my lettero fOctober26 , v i . thatrotationa l

absolute lydiscarded andw emustha e

o lutionandvacuouscores. Sonotto

o f c or e le s s v o r te w o r ( ' V i b ra t io n s of a

P ro c. R .S .E . M ar ch 1 1 88 0 , H ic s P ap er ' O n

mallV ibrationsofaHollow V orte , '

1884 , willbethebeginningof the

erandmatter if it ise ertobeatheory .

ossinglinesofvorte co lumn isimpossible

utispossiblew ithv acuouscoresandpure ly

oundthem.Theaccompanyingdiagram

pla inbyanil lustration. Itshow stheshape

rica lv acuousv orte co lumnasdisturbed

f i edinaplaneperpendiculartothea is

a ingirrotationa lcirculationthroughitse lf .

acuum thespaceoneachside li uid

essectionof thetore. Thecur esrepresentingtheboundaryof thev orte areca lculatedtogi euniform

erthew holesurfaceof theho llow core .

antof thev elocitiesduetothecirculation

e andtocirculationthroughthetore . The

erseproportiontodistancef romthea is

The latterisappro imate lyparalle lto

erseproportiontothecubeof thedistance

rcularcrosssection l i eananchorring.



s s_ u s e # p d



orticesisstable butitsquasi- rigidity

siona lmotion w ithoutchangeofv o lume

edingly small andthecorrespondinglaminar

ingly sluggishincomparisonw iththetensile

orrespondingwa e- e locity w hichweshould

rmotioninplanes paralleltotheplane

erygreatnumberofplanesina lldirectionsasmanyw ithinanangleof1~ ofanyoneplane asw ithin1~ of

eadistributionofstraight v orte

presentedinf ig. 2 , perpendiculartoe ery

res beingthinenough theymaybe

nooneofwhich intersectsanyother.

ev orticeswillproducedisturbances

w hichw esupposedthemgi en and

ationsf rome actlycircularf igure in

dtherewillbesluggish motionsofthe

lplaced soastofulfil adefinitecondition

enif thisdefinitecondit ionisnote actly

asi- rigidity andcorrespondingve locityof

edium thusk inetica lly constituted w ill

mwhat theywouldbeifthe v ortices

eabsolutelysteadymotionforthe

themedium.

uslyconsideringtheef fecto f f reevorte

esamongthevorte co lumnsof thistensile

ggestedforcoredvorticesattheendofyour

6 1889 to lNature . Itw illbean

dynamicalq uestion thoughitseemsto

tletowardse pla ininguni ersa l

propertyofmatter soyoumay imagine

chemistryandelectro-magnetism.



s s_ u s e # p d



F IC IENC YOF SA ILS W INDMILLS SC R EWPR OPELLERSINWA TER A NDA IR A NDA ER OPLA NES.

o l . L. 1 8 94 p . 4 25 .

wee , onflyingmachines inthe

not forw antof t ime carriedsofaras

rica lresultso fobser ationputbefore

m thattheresistanceof thea iraga inst

omo eatsi tymilesanhourthrough

inedtotheplaneataslopeofaboutone

eaboutfifty-threetimesas greatasthe

ld" theoretica l ( ) f o rmula andsomethingli e f i eortentimes thatcalculatedf romaformulawritten

ordR ay le igh asf romapre iouscommunicationtotheB rit ishA ssociationatitsGlasgow meetingin

rewasnov a lidity e enforrough

nanyof the" theoretica l in estigations

wwildlytheyall fallshortofthetruth

ehadopportunity inthe lastfew days

aminesomeof theobser ationa lresultsw hich

introductiontohispaper. Ontheother

erdoubtedbutthatthetruetheoryw astobe

tcon ersationallybyWill iamF roude

ch thoughIdonotk now of itsha ing

hitherto isclearlyandterselye pressed

hichI q uotefromatype-writtencopy

rMa im ofhispaperof lastw ee : Thead antagesarisingf romdri ingtheaeroplanesonto

w hichhasnotbeendisturbed isclearly

ments.

es seefootnote p. 219infra .



s s_ u s e # p d



p le I h a e a t l as t I b e li e e s u cc e ed e d

meapproachtoaccuracy theforcere uired

ow rectangularplanemo ingthroughthea ir

e locity V , inadirectionperpendicularto

clinedatanysmallangle i to itsbreadth a .

etobeabletocommunicatetothe

intimeforpublicationinitsne tO ctober

ethe in estigation includingconsideration

andproof thatit iso fcomparati e ly small

muchmorethan1/ 10 or1/ 20 o faradian

somepractica lly smooth rea l so lidmateria l.

stheresult w iths in- resistanceneglected:Theresultantforce( perpendicular therefore totheplane is

w h ic h i s v r c o s i /s i n i ti m es ( o r f or t h e ca s e of

ndredtimes , theo ldmisca lled" theoretical

merica lfactorw as2insteadof1.



s s_ u s e # p d



07 )

NC E O F A F L U I D T O A P L A NE K E P T M O V I N G

REC TIO NINC LINEDTO ITA TA SMA LLA NGLE.

ca l M ag a i n e V o l . x x x v i I I. 1 8 94 p p. 4 0 9 â € ” 4 1 .

ity i itsinclinationtotheplane and

nandperpendiculartotheplane. Weha e

i n i.

ingbodytobenot anidealinfinitely

f f initethic nessverysmallincomparison

andha ingitsedgese erywheresmoothly

iscidandincompressible andthe

fectlyunyielding themotionproduced

byanymotiongi entothedisc isdeterminate ly theuni uemotionofw hichtheenergy islessthanthat

eto thefluidwiththegi enmotion

thedisctobev ery thin andtherefore

te erypo into f itsedgetobeverygreat:

nessat whichthepropositioncould

til lho ldsinthe idea lcaseofaninf inite ly

andits boundaryfulfiltheidealconditionsofthe enunciation.

ry f luidhassomedegreeofv iscous

hape andanyv iscosityhow e ersmall

erfectincompressibil ityof thef luidandun-

ary w ouldpre entthe inf inite lygreat

f thediscw hichtheuni ueminimumenergyso lutiongi esw henthediscisinf inite ly thin andw ould

ancein themotionofthefluid that

nof thediscwouldprobablybev ery

ertheactua lv a lueof thev iscosity ifnot

ththe v elocityofthediscmultiplied



s s_ u s e # p d



atureof theboundaryof itsarea. No

hashithertobeenmadetow ardsacomplete

nycaseofthis problem orindeedofthe

sapethroughav iscousf luid e ceptw hen

lso lutionsfortheglobeandcircularcy linder

tsconfigurationisthe sameasitwould

w andw hentherefore theve locityof

se ua lto andinthesamedirectionas

lacementofanelastic solidwhena

sheldin apositioninfinitesimallydisplacedfromitspositionof e uilibrium inthemannertranslationally

dingtothetranslationalandrotational

rigidbody inthefluid.

guidedby theteachingofWill iam

ntinuedcommunicationofmomentumto

of forcetok eepaso lidmo ingw ith

ocity throughit thatanappro imate

tance w hichisthesub ectof thepresent

robablybefoundbythefollowingmethod

~ 9 w hichIv enturetogi easaguess

mathematicalin estigation.

nitethic ness how e ersmall mo ing

ssible li uidw ithinanuny ieldingboundary

hin ingonlyof theu-componentof themotion

of~ 1 le tEandE denotethef rontand

respecti e ly . Imaginenow insteadof

ary ingso liddiscthroughthef luid that

E by rigidif icationandaccretionof the

meltsaw ay fromE by li uefactionof the

me3t thee tentof theaccretionin

fthe v -componentofthemotionof

houtdiminutionduringthisaccretion

l to ( r I - I / t m u st b e a pp l ie d f ro m wi t ho u t

Idenotingthe impulsi e forcew hich

ethev -componentv e locity totheunaugmenteddisc andI thatre uiredtogi ethesamev elocity to

pointofapplicationofthe force

orthesteady infinite ly slow motionofav iscousf luidare

e uilibriumofanelasticsolid. SeeMathematicaland

. T ho m so n , V o l . II i . Ar t . c i . ~ ~ 1 7 1 8 .



s s_ u s e # p d



ST A NC E O F A F L U I D T O A P L A N E

hatof theresultanto f impulsesIand- I

ntresof inertia oftheaugmenteddisc

crespecti ely.

rigidity byli uefactionofany

e ly small o fmattero f thediscatE

ousapplicationof force topre entchange

sidualsolid.Thecontinuedgradual

resupposingperformed lea esaHelmholt

f f in i te s l ip g r ow i ng o u t in t h e li u i d b e hi n d E ,

ortionsofwhicharenot easilyfollowedin

tisintheformofapoc etofw hichthe

dtothesolid disc.Thespaceenclosed

f il ledby the li uidw hichw asso lid.

ndlongerbyga inof li u idf romthe

ontof it andprobablya lsoby itsrear

arther faraw ay inthew a eof thedisc.

a f terha ingbeenperformedduringa

aprocessesof~ ~ 5 6arediscontinued and

e ualandsim ilartotheorigina ldisc but

hroughaspacee ua ltouT isle f tw ith

oughthef luidmainta ined. Thepoc etof

fartherandfartherbehindthedisc. Its

pedby theso lid w il lshrin f romitsoriginal

eofE andw illbecomealw ayssmallerand

eysmallinany f initetime. Thenec o f

eof thediscw illbecomenarrowerand

epoc etw illbedraw noutlongerand

hroughallt ime thef luidw hichwasso lid

surfaceof finiteslip orHelmholt

r om t h es u rr o un d in g f lu i d e c e pt o e r t he e e r

sc w hichstopsthemouthof thepoc et.

rotationaloutsidethepoc et and

eepthesoliddiscmo ingw ithits

ndwithnoothermotionwhetherrotational

ecessary toapply forceto it. B utthisforce

andappro imatestoz ero asthevorte * Ica llthe" hydrauliccentreof inertia o famasslessrigiddiscimmersedin

itmustbestruc perpendicularlybyanimpulse togi e

tion.

14



s s_ u s e # p d



r andthemotionofthe fluidinthe

appro imatesmoreandmorenearlyto

euni ueirrotationalmotiondueto

ughthe fluid.

~ ~ 5 6 7 b ee n on s ur e gr ou nd a nd

rously true notonly fora" disc o fany

fany thic nesshowe ersmall but

pe dea ltwithaccordingto~ 5

f luidisin iscidandincompressible and

Myhypothesis or" guess ( ~ 4 , w hich

f thepresentpaper isthatdefaultf rom

tofallthesethree conditionswould

eptmo ingwithuniformtranslational

1 , r e u i re t h e co n ti n ue d a pp l ic a ti o n to i t o f fo r ce

andposit ionby~ 5 pro idedv bev ery

outw ithgreateaseforthecaseofa

he length 1 isv erygreatincomparison

orthiscase by thew ell- now nhydro ineticsofane ll ipso idore llipticcylindermo ingtranslationa lly

ssible f luidofunitdensity w eha e

tionof~ 5

2 I .

7 r al u ;

intofapplicationofthis forcefromthe

is 4a.

hetica lresult w ithobser ation in

udeof theforceandits pointof

pe form thesub ectofa futurecommunication.



s s_ u s e # p d



F A H E TE R O G E NE O U S L I Q U I D C O M M EN C IN G

G IV E N M O T I O N O F I T S B O U N D A R Y.

gsof theR oyalSocietyofEdinbw rgh V o l. xx I.

id forbre ity todenoteanincompressible

scid butin iscidunlessthecontrary ise pressly

i u i d v i s ci d o r in i s ci d b e in g g i e n

ingv esselo fanyshape w hethersimplyor

tanymotionbesuddenlyproducedin

ry orthroughouttheboundary sub ect

dtionofunchangingvo lume. E ery

linstantaneouslycommencemo ingw ith

yandinthe determinatedirection such

fthewholeis lessthanthatofany other

couldha ew iththegi enmotionof its

tionisalso trueforanincompressible

( andforthe idea l" ether o fP roc.

andA rt. xcI . v o l. III. o fmyC ollected

lPapers .Thetruthof theproposition

li uidisvery importantinpractica l

pleof itsapplicationto in iscidand

sticso lid considerane lasticj e lly standing

ande ualbul so fw aterandofanin iscid

nMathematica lJ ournal F eb. 1849. Thisisonlya

k inetictheoremforanymaterialsystemwhate er

a lSociety Edinburgh A pril6 186 , w ithoutproof

6 p . 1 14 a n d pr o e d i n Th o ms o n an d T ai t s N a tu r al

w ithse era le amples. Mutualforcesbetw eentheconta iningv esse landthe li uidore lasticso lid suchasarecalledintoplayby

hesi ity ( orresistancetoslidingbetw eenso lidandso lid ,

sion anddonotenterintothee uationsusedinthe



s s_ u s e # p d



e u a l an d s im i la r t o it . G i e e u a l su d de n

ningv essels:theinstantaneousmotions

bstancesw illbethesame. Ta e asa

eof re o lutionw ithitsa isv erticalforthe

dletthegi enmotionberotationroundthis

edandafterwardsmaintainedwithuniform

itia lk ineticenergyw illbezeroforeach

Thein iscidli uidw illremainfore er

c uiremotionaccordingtotheF ourier

k nowsomethingforthiscaseby

tofgi inganappro imatelyuniform

ertica la istoacupof tea init ia llyat

uire laminarw a emotionproceeding

y.B utinthepresent communication

othecaseof in iscidli uid.

ution oftheminimumproblemthus

undingsurface issimplycontinuous is

ionoftheli uidisirrotational.

tbe irrotationaltisindeedob ious

mpulsi epressurebywhichany

etinmotionise erywhereperpendicular

and thecontiguousmatteraroundit

mentofmomentumroundany

ricalportion largeorsmall isz ero . B ut

otionofe erysphericalportionofthe

nethemotionwithin asimplycontinuous

tatedmotion isnotob iousw ithoutmathematica lin estigation.

simplycontinuous ormultiplycontinuous irrotationalitysufficestodeterminethemotionproduced

eproduced f romrestbyagi enmotion

uidactedonbynobodily force or

i t y f o r e a m pl e a s c ou l d no t m o e i t

ed themotionstartedf romrestbyany

ary remains a lw aysirrotationa l asw e

NaturalPhilosophy ~ 3 12.

suchthatthemomentofmomentumofe eryspherica l

isz erorounde erydiameter.



s s_ u s e # p d



hetends toha efaithinall assertions

nce.

y isanenclosingv esselo fany rea lmaterial

erfectly rigidnorperfectlye lastic andif

f tto itse lf underthe inf luenceofgra ity

perfectly in iscid w il l loseenergycontinua lly

heconta iningv essel andw illcome

econfigurationof stablee uilibrium

ensityhori onta landincreasingdensity

dt io n s as i n ( 3 ) , b u t n o gr a i t y t h e

estwillbe infinitelyfinemi ture

o f e u a l de n si t y th r ou g ho u t . C on s id e r f o r

ogeneousli uidsofdif ferentdensit iesf i ll ingthe

nglehomogeneousli uidnotf i ll ingit. A san

ttlehalf fullo fw ater andsha eitv io lently .

hew holebottle fullo fami tureof f ine

mogeneousthroughout.Thin whatthe

erenogra ity andif thewaterandair

ott lesha enasgentlyasyouplease and

cuuminplaceof theair or if f o ra ir

uidofdensitydifferentfromthat of



s s_ u s e # p d



E O F D I SC O N T I NU I T Y O F F L U I D M O T I O N I N

THER ESISTANC EA GA INSTA SOLIDMO V ING

D* .

l . L. 1 89 4 p p. 5 24 5 49 5 7 , 5 97 .

discontinuity " thatistosay f inite

ntw osidesofasurface inaf luid w ould

dincompressiblefluidwerecausedto

arigidsolid withnov acantspacebetween

e li e e f i rs t g i e n b y St o e s i n 18 4 7t .

owwell- nowndynamicaltheorem

iscidf luidinit ia llyatrest andsetin

dto itsboundary ac uirestheuni ue

ughoutitsmass ofwhichthek inetic

anyothermotionofthe fluidwiththe

ry.

dforthe formationofasurfaceof

dfluid wastheinfinitelygreatv elocity

andthecorrespondingnegati e - infinite

euni ueso lution unlessthef luidisa llow ed

ntactwiththesolid. Thisanin iscid

nlywoulddo unlessthepressureofthe

e erywheree ceptattheedge. In

erygreatnegati epressurearisingf rom

ncationsformedthesub ectofapro longedplay fulcontro ersy

ndhisintimatef riendSirGeorgeSto es inaseriesof letters

r ed.

o l . i. p p . 3 1 0 3 1 1 .



s s_ u s e # p d



afluid flowingroundacorneris always

f threedefa lcationsf romourideal: I V iscosityof thef luid pre entingthee ceedinggreatnessof thev e locity .

yof thef luid.

theouterboundaryof thef luid.

isinmanypracticalcaseslarge lyoperati e

t( II isa lso large lyoperati e insome

suchasthewhistlingofastrongw ind

rnerorthroughachin theblow ing

theenmbouchureofanorgan-pipe and

geo letoro fasmall" w histle andthe

tubeorahole inthesideof atube

etosound.

i s l ar g el y o pe r at i e a n d ( I I b u t li t tl e

mostcommonoccurrenceintheflowof

muchofthefoamseennear thesides

ew steamergoingatahighspeedthrough

ueto" v acuum behindedgesandroughnessescausingdisso l eda irtobee tractedf romthewater. A

hdiameter and1/10ofaninch thic

ulytothefigureof anoblateellipsoidof

useavacuum* tobeformedallrounditsedge

mallav e locityasIfootpersecondunder

han6 feet ifw aterw ere in iscid: and

tonw ould onthesamesupposit ion be

nov acuum andw ouldbee actly in

ueminimum energysolutiont.

ntespacev acatedbyw ater.

ydro ineticsofthemotionof anellipsoidthroughan

f luid originatedbyGreen w hof irstga etheso lutionfor

motionof thee llipso id w ek now that if0denotesthe

tothesurface atanypointandthe a isofanoblate

ofw hichthee uatoria landpolarsemia esarea b the

wngo erthispo into f thesurface is

udatgreatdistancesf romtheso lidisV , andinpara lle l

e ldf i edinthef luid w ithitsa ispara lle lto these lines.



s s_ u s e # p d



NE O F F I N IT E S LI P I N F L U I D M O T I O N 2 1 7

hef luidacrossthee uatoris6 ' 7feet

ocityacrosseachof thetw oparalle lcircles

nches( theradiusof thee uatorbe ing

persecond.

y rapidchangeofshapeof thef luid

toria lz onebetw eenthesecircles w ith

augmentingf rom1footpersecondto6 ' 7

ncingo eradistanceof lessthan' 85of

moneof thesmallcirclestothee uator

m6 ' 7to1f romthee uatortotheother

tionofasecondof timew ould if thef luid

i u i d g i e r i se t h ro u gh v i s c o s it y t o

thema imum v e locity andcausing

themotionofthewaterto differgreatly

energysolution notonlynearthe

e o r o e r t he r e ar s i de o f t he d i sc b u t o e r

thoughnodoubtmuchmoreontherear

thanonthef rontsideandinthef luid

atlessdepthsthan6 feet ha e

downthema imumvelocity andit is

0or20feetagreaterve locity than1foot

uiredtoma ev acuumroundthee uator

meterandthe1/2000of aninchradius

llipticmeridiona lsectiongi esit. B utit

theremust bemuchformingofv acuum

actionofa irandrisingofbubblestothe

whatsharpcornersandroughnesses of

dinary ironsa il ingshiporsteamer go ing

ek nots( thatis 20f t. persecond .

whichvacuumisformedatanedgeofa

iscidincompressible f luid underpressure

atdistances fromthesolid asuccession

formula wereduceitto 200/7r.( V sin0 appro imatelywithin

n g si n 0 = 1 a n d V = l f o o t p er s e co n d w e fi n d 6 - 7 f ee t p er

acrossthee uator. Hencethegra itationa lheadcorrespondingtothe" negati e -pressure is( 6 -72-12 / 64-4 orv eryappro imately

e s t he s t at e me n t in t h e te t .



s s_ u s e # p d



restingpieceofmathematicalhydro inetics

ntinuationofthepresentarticlein which

f f luidmotion e tendingfarandwide

sinmanyscientif icpapersandte tboo ssinceSto es inf initesimalrif tstartedit in1847 w illbe

ngdiagram( f ig. 1 i l lustratesthe

e inquestion toadisck eptmo ing

A

aconstantve locity V , perpendicularto

ptiontow hichIob ectasbe inginconsistentw ithhydrodynamics andvery farf romanyappro imationtothetruthforanin iscidincompressible f luidinany



s s_ u s e # p d




tterlyatvariancew ithobser ationofdiscs

s causedtomo ethroughw ater is that

representedbythetwocontinuous

ande tendingindef initely rearwards there

nuity ontheoutsideofw hichthew ater

hedisc w ithv e locityV , andonthe inside

ssmassof " deadw ater fo llow ingclose

stancyofthev elocityontheoutside

discontinuityentailsfor theinsidea

thereforeq uiescencere lati e ly tothedisc

deadw ater. How couldsuchastateof

ndw hatisit inrespecttorear are

ggestto theteachersofthedoctrine

go inginforane aminationinhydro inetics Ineednottry toansw er.

u pp o si n g th e m ot i on o f t he d i sc t o h a e

time t ago andconsideringthe

~ 9 f o r fi n it e ne s s of i t s wa e l e t ab b d b e

dtherear andbeyondoneside o f the

ssonlythroughwaternotsensiblydisturbed.

nitecaseofmotiontodea lw ith instead

teoneof~ 11. Letustry if it ispossible

efrom theedge andfromthediscon

ouldbee enappro imate ly ifnotrigorously

andindicatedby thediagram.

e l oc i ty a t a ny p o in t i n th e a i s A a a t

rearwards.Drawedperpendiculartothe

re lati e ly tothediscsupposedatrest.

eedis0 , , , , , , db , V x db , , , ba , 0

, , , , A e , O , b yh yp ot he si s.

pressionofpracticalhydraulics adoptedby theEnglish

finiteslipbetweentwopartsofa homogeneousfluid to

eati e ly tothedisc.

( T h om s on , T r an s . R. S . E .1 8 69 .



s s_ u s e # p d



on intheclosedpo lygonedbaA e

ationinthesamecircuit a tatime

T w henthe linebahasmo edtotheposit ion

d , w e ha e

f o r th e l at e r ti m e t + r t h e v e l oc i ty i n A a a t

e circulationinedbaAetgainsin time

oremof" circulation " + must bee ualto

imeT o fa llthevorte -sheetinits

rdingtothe statementof~ 11.Hence

0

- ( v ' - v ) dy.

ow thatthef luidhasonlycontinuous

hafinitespaceall roundeachofthe

A a n d al l r ou n d Ae e c e pt t h e sp a ce o c cu p ie d

eyonditsf rontside w eha e forthe

smotion re lati e ly tothedisc

z , t ,

ocity-potentialofthemotionrelati eto

l round:andweha ealongAa

0 y 0 t .

o f~ 14becomes

0 0 0 t + r } - { ( 0 0 0 t } .

culationinabb a isz ero andthereforethecirculationin

to t h at i n e db a Ae .

sthatthere isasmoothstreamlinef romtheneighbourhoodof

idea llthev orte motion foronly thenisthecirculationin

n " T r an s . R. S . E .1 8 69 .



s s_ u s e # p d



NE O F F I N IT E S LI P I N F L U I D M O T I O N 2 2

f inite ly small

t .

omttot+ T therehasbeen according

n~ 11 agrow thofv orte -sheetf rom

eingthemeanbetw eenthev e locit iesof the

ndthecirculation perlength1of the

Hencethev orte -circulationof the

intimeT byV Tx V : andtherefore

2 .

testhepressureof thef luidatgreat

e locity re lati e tothediscisV , andpthe

erearsideofthe disc beingthesame

ha e bye lementaryhydro inetics

o , t ,

thef luidate erypo into f therearside

rdingtotheassumptionof " deadwater.

asthepressureontherearsidegi enby

ionofanendlesse erbroadeningw a e

r o e s t ha t o ur s u bs t it u ti o n ( ~ 1 ) o f a f in i te

once i ablypossibleastheconse uence

onatsomef initetime t ago instead

nf igurationdescribedin~ 11 doesnota lter

deofthedisc.

tionofthe fluidforsomefinite

nbothitssides thesame orv ery

m asthatdescribedin~ 11 theforcethat

pitmo inguniformlywouldbethesame

y thesame asthatcalculatedbyLord

ofthefluid supposedtobewhollyas

enschaf tl icheA bhandlungen V o l. i. f ooto fp. 151.



s s_ u s e # p d



ha ew eforsupposingthev e locity

onthef rontsideof thedisc tobe

imate lye ua ltotheundisturbedvelocity

atdistancesf romthedisc NonethatIcan

dprobablethatitis inrealitymuch

nweconsiderthat w ithin iscidincompressible f luidinanuny ie ldingouterboundary theve locity in

4 ise ua ltoV ate ensofarf romthe

a n d in c re a se s f ro m V t o 6 ' 7 x V b e tw e en

e andtheedgewithits 1/2000ofan

e.

deadw ater incontactw iththe

whichthedoctrineof discontinuity

erealityandyouw illseethew aterinthe

ye erywheree ceptatthev erycentreof

dyingroundfromtheedgeand

yclosea longtherearsurface o f tenIbe lie e

city thanV , butw ithnosteadiness on

rbulentunsteadinessutterlyunli e the

erallyassumedinthedoctrineof discontinuity.

sa fe lyconcludethatonthef rontside

ss thanthatcalculatedbyRayleigh.

stanceis partiallycompensatedoris

dminutionofpressureontherear ismore

mtheoryalone inaproblemofmotion

yondour powersofcalculation:butwe

belie e bye periment. R ay le igh s

istancee periencedbyaninf inite ly thin

bytwoparallel straightedges when

ganin iscidincompressible f luid w ith

, inadirectionperpendiculartotheedgesand

eplane gi esaforcecuttingtheplane

cefromits middlee ualto

esfortheamountof thisforce ingra itation



s s_ u s e # p d



NE O F F I N I T E SL I P I NF L U I D M O T I O N 2 2 5

aofonesideof theblade andPthe

fluidofunitcross-sectionalarea andof

htf romw hichabodymustfalltoac uire

.

~ 11 onw hichthisin estigationis

cityof f luidmotion re lati e ly tothedisc

. Itgi esv e locity reachingthisva lue

ade andatthesupposedsurfaceof

thef luidatinf initedistancesa llrounde cept

eof " deadw ater w herethev e locity

pressuree ua lto Ia llthroughthe" dead

sit increasethroughthemo ingf luid f romII

andatthe" surfaceofdiscontinuity toa

Patta inedatthew ater-shedlineof the

difV bee ensogreatas120feet

e locityofsound * P w ouldbeonly

dingaugmentationofdensitycould

angeofthe motionfromthatassumed:

sin estigationa irmayberegardedas

ev elocityofthedisc isanythingless

hisformulafortheresistance by

carefule perimentsmadebyDinest

discsandbladesmo edthroughitat

70statutemilesperhour( 59to10 feet

fornormalincidencetheresistanceaga inst

mo ingthroughairatmB rit ishstatutemiles

to ' 0029nm2ofapoundw eight.

ecificgra ityoftheair as1/800 gi es

f~ 21

uareplateofareaA . A tthefooto f

ne1890 Dinessaysthathef indsthe

blade tobemorethan20 percent.

eplate . F ora bladewemaythereforeta e

H w hereHis" theheightof thehomogeneousatmosphere.

15



s s_ u s e # p d



gtoDines e periments. Thisis2 94

latedf romR ay le igh sformula( 21

moreandmoreobli ue thediscrepancy

us f romcur esgi enbyDines( p. 256

eigh sresults If indthenormalresistance

ughairinadirectioninclined30~ to its

imesthatgi enbyR ayle igh sformula . A ndby

es cur eatthepo intinw hichitcuts

e If indthat forv erysmallv a luesof i it

urtimesthev alueoftheforcegi en

orv erysmallv a luesof i w hichis

doublethatgi enbymycon ectura l

ust3 0 p. 426 andPhil. Mag. October

a l ue s o f i w hi c h i s

er mere lycon ectura l andIw asinclined

siderablyunder-estimatetheforce*. That

isperhapsmadeprobablebyits somewhat

es becausetheblade inhise periments

o faninchthic inthemiddlew ith

Aninfinitelythinbladewouldprobably

istances ata llangles andespecia llyat

the wind.

eha eherebeendoubled inadditiontootherslight

outtheappro imateagreementwhichwasfoundby

rinformationregardingtheresultso f recentin estigation

boratory DrT . E. Stantonw rites( O ct. 7 1909 as

diagramanne ed. " Theva lueofDines coeff icientq uoted

remar ablyw ellw ithourresultsherew henaccountista en

si eof theplate asyouw illseef romtheencloseddiagram.

otalresistanceperunit areawithsi eisentirelydue to

ctatthe bac oftheplateasthe dimensionsincrease.

resonlongnarrow platesandoninclinedplates our

eementw iththoseofDines andonourin estigatingthe



s s_ u s e # p d




edemonstrationthatthedoctrineof

rf romanappro imationtothetruth is

edingly interestingandinstructi emanner by

f thepressuresonthetw osidesofadisc

lati ew indof60statutemilesperhour

roducedbycarry ingitroundattheendof the

machine. Theobser ationsw eredescribed

RoyalMeteorologicalSocietyin May

neof thesameyear intheR oya lSociety

redto hestatestheresults w hichare

ontsideanaugmentationofpressure

arsidea diminutionofpressure

by1 82and' 89inchesofw ater w erefound.

sofair ofdensity1/800ofthatof water

1211and59~ feet. Theformerisina lmost

rigorousmathematica ltheory foranin iscid

hichgi es882/64 4 or1201feetforthe

efoundthatthee cessinthetota lresistanceo erthat

sformulaw as asinthecaseofnormalimpingement due

eddiesontheleewardside.

eplates.

i

10

n s ma d e by M . Ei f fe l o n fa l li n g pl a te s .. , , , , , a t N .P .L . i n a cu r re n t of a i r. , , i n t he w i nd . , , , , , b y M r Di n es o n w hi r li n g ta b le .



s s_ u s e # p d



YNA MIC S [ 25

epressureatthe water-shedpointor

nshapemo ingthroughitattherateof

er showsthatthereisa " suction

dev erynearlye ua ltoha lf theaugmentationofpressureonthef ront insteadof therebe ingneither

essureastaughtin thedoctrineofdiscontinuity

t o t/

r/////////////////////

ng d i ag r am s ( 2 3 , 4 5 r e pr e se n t se e r al

of discontinuityinthemotionofan

tracti etow ritersonmathematica lhydro



s s_ u s e # p d



NE O F F I N I T E SL I P I NF L U I D M O T I O N 2 2 9

sentedinF ig. 1( w hetherasitsstands or

incidence becauseeachisinstantly so luble

aysis andtheydonot l i e it inthe

constituteillustrationsofthebeautiful

findingsurfacesofconstantfluidv elocity

surfacesa longw hichthev e locity isnot

yHelmholt * , de e lopedinamathematica lly

byKirchhof f t andva lidlyappliedtothe

ontracta byR ay le igh+ .

t( notnecessarilyo fcircularcross-section

harpedge intoavery largev o lumeof

asthato f the j et isrepresentedinF ig. 2.

sideredbyHelmholt ~ , bothforthe

siblefluidandforreal waterorrealair.

o rbelie ingthat w ithrealw aterorreal

omthemouthasgreatasse era lt imesthe

rthe leastdiameter if it isnoto fcircular

undingf luidisnearlyatrest andthe j et

thek indofmotionithad inpassing

forethattheefflu isnearlythesame

esthesame itw ouldbe if theatmosphere

hargedwereinertia-less.Thisconclusion

nceinpracticalhydraulics hasbeen

perimentsmadeeightyearsagointhe

ni ersityofGlasgow by tw oyoung

a y MrC appsandthe lateMrHew es.

tedandconf irmedbyothere perimenters.

stapplicationofthedoctrineof discontinuitytothetheoryofthe resistanceoffluidstosolidsmo ing

entedinF ig. 3, andtheresult isno

thiscase re uiringnoca lculation might

f thee tremew rongnessof thedoctrine in

eoffluidsagainstsolids mo ingthrough

stancein thecaserepresentedby

heassumptionofaw a eof " deadw ater

eA bhandlungen " V o l. I. pp. 15 -156.

M at h em at i sc h e Ph y si , V o l . x x I .

namics " Phil. Mag. 1876 secondhalf -year.

V o l . i. p p . 15 2 -1 5 .



s s_ u s e # p d



D Y NA M IC S [ 2 5

ure 1I asthedistantandnearw aterf low ing

followsimmediatelyfromaneasily

tatedin thecombinedmeetingofSections

. , i n O x f o rd l a st A u gu s t t o t he e f fe c t th a t th e

thepressureoneachof theends E E , in

w ha t e e r t he i r sh a pe s a n d w h et h er " b o w o r " s t er n "

dstangentia lly inacy lindric" mid-body

egreatesttrans ersediameterofthe

A w hereA istheareaof thecross-sectionof

id.

presenttw ovarietiesofacasew holly

ableendlessnessofF ig. 1 andcarefully

nsiblebyholdersofthe doctrineofdiscontinuityifit hasanydefensibilityatall.I v enturetolea eit

ration.



s s_ u s e # p d



ES .

RO R INLA PLA CE STHEOR Y

calMaga ine V o l. L. 1875 pp. 227-242.

nT idesandWa esintheEncyclopediaMetropolitana gi esav ersionofLaplace stheoryof the

btedlygreatmerit ofbeingfreedfrom

pcationsof " Laplace scoef ficients " or as

monlyca lled " Spherica lharmonics " by

rattempted notq uitesuccessfully to

ngintoaccountthea lterationofgra ity

ceofthesurfaceofthe seailthesolution

ons.

temptforeach ofthe" three

M e ca n i u e C el e st e L i . I V . a r t s . 5 7 9 ,

inthecourseofw or ingoutthe

uations( art. 3) ] by theassertionthat

e t o b e ra t io n al f u nc t io n s of p a n d V / 1 U 2 " 2 t h at

arly thew holechapter( Li . IV . chap. i o f

ideof t idaltheorywasunderta enbyaC ommitteeof the

hichpublishedvariousreports( B . A . R eports 1868 70 71

epracticalmethodsofharmonicanaly siso f t idesandtheresults

ocean. O neof the longerof these( B rit. A ssoc. R eport 1872

draw nupbyMrE. R obertsunderthedirectionof theC ommittee : thene t( B rit . A ssoc. R eport 1876 pp. 275- 07 w as" draw nupby

se uently thedirectionof thisw or w asta eno ermainly

f . Thomson&amp Ta it sNat. Phil. ed. 2andSirGeorge

entif icPapers V o l. II. ( w hichincludefurtherB . A . R eports

boo onTheT ides.

nteddrew attentionafreshtotheLaplaciantheory

mpro edandde elopedbyv ariousw riters includingDarwin

dinparticularbyS. S. Houghintw omemoirsinPhil. T rans.

p . 2 0 1 a n d V o l . 1 9 1 A ( 1 8 98 , p . 1 9 : c f. L a mb s H y dr o dy n am i cs

ddresses v o l. III. Na igation 1891 aB rit ishA ssociationlecture( Southampton 1882 on" TheT ides isreprinted pp. 1 9-190with

C D E o f w hi c h B , C a r e pa p er s ( B r i t . A ss o c. D u bl i n 1 8 7 8 o n

ra itso fDo erontheT idesintheB rit ishC hannelandthe

ntheT idesof theSouthernHemisphereandof theMediterranean " the latterincon unctionw ithC apt. E ans w hileDisa" S etchofP roposedP lanofP rocedure inT ida lObser ationandA nalysis" f romB rit. A ssoc.

68.



s s_ u s e # p d



ES

de otedtothedynamica ltheoryof the

festhefo llow ingstatementwithw hichA iry

s " a r t . ( 6 6 ] i n t r od u ce s h is o w n v e r si o n of

w ouldbeuse lesstoof ferthistheory inthesameshapein

nit f o rtheparto f theMe cani ueC eleste

ofTidesis perhapsonthewholemore

arto f thesamee tentinthatw or . We

aforme ui a lenttoLaplace s andindeed

apersonfamiliarwiththe latterwill

o f thesuccessi esteps. Theresultsat

ethesameasthoseofLaplace.

stinA iry streatisethroughthe

armonicanalysisis Laplace scomplete

rotationande ualdepthof thesea

ntheearth srotationista eninto

ea iso fune ua ldepth thedifferentia l

ta esaformaltogetherunsuitedforthe

armonics ; andA iry sin estigationis

Laplace s e ceptinthe j udiciousomission

ptsreferredtoabo e.

sso lutionforthesemi-diurna lt idew ith

uetothechangeof f igureof thew ater

A irypo intsoutwhathebelie edtobe

at a f tercorrectingit itw as" needlessto

snumerica lca lculationsof theheightsof

andhisinferencesasto thelatitude

amp c. fa llto theground. WhenIf irst

nyearsagoonboardthe ' GreatEastern '

rrectionofLaplace but onthe

elfthatLaplacewasq uiteright.Not

eC elesteathand Isetthesub ectaside

oreturnto itforthesecondvo lumeof

NaturalPhilosophy ' w iththef irstvo lume

ed.

ybeenrecalled toitbyreading

R esearches ' constitutinganappendi

re lto the ' UnitedStatesCoast-Sur ey

ghsubse uently succeededinintroducingharmonicana ly sis



s s_ u s e # p d



GEDER R OR INLA PLA C E STHEOR Y

hefo llow ingpassagereferringtoLaplace s

ltides: Theresultsshowthatthe formwhichthesurfaceofthe

iffersv erymuchfromthatofa prolate

isinthedirection ofthedisturbing

ndepthsof theoceanthetidesat the

low w aterta ingplaceundertheattracting

how e er thetidesw erefoundtobe

nde eninthecasesinw hichtheyare

ortheyw erefoundtobedirecttowardthe

ntly there isa latitude insuchcaseswhere

ce how e er fa iledto interpretcorrectly

on sothatthenumericalresultswhich

tassumeddepthsoftheocean are

genera lresultsj uststatedarereadily

ingfailedtoseethe indeterminate

headoptedasingularandunwarranted

hev alueofaconstantwhich isentirely

f rict ion butw hichv anishesinthe

ersmall. Thiso ersighto fLaplaceand

hisconstantweresubse uentlypointed

ngularandunw arrantedprinciple thus

q uisitely subtlemethodbyw hichit

minedaconstantwhichis notarbitrary

cannotbemorethaninfinitesimally

riction. F errelfurthere tendsto

orthe" diurna lt ide theob ectionof

Airyhadraisedonlyagainsthis

rnal andhefollowsAiryin an

nbyLaplace forthe" long-periodtide "

ceofdeterminateness( strangelyinconsistentwiththeindeterminatenessassertedofthesolutionsfor

iurna l isproducedby the inad ertent

thetrueva lueofw hichistobedeterminedbyaproperapplicationofLaplace smethod. Withthese

notwaittwoor threeyearsmoreforthe

homsonandTait sNaturalPhilosophy to

ess butmustspea outonthesub ect

e t( O ctober Numberof thePhil. Mag. entit led" Note

eF irstSpeciesinLaplace sTheoryof theT ides. [ Inf ra.



s s_ u s e # p d




dA iryassume

4 + ... + K 2 x 2 + K 2 + 2 2 + 2 + & amp c... 3 )

entia le uation( 2 . Then bye uating

hele f t-handmemberto -8H itscoeff icient

uatingtoz erothecoef f iciento fc2 +4 fora ll

oo they f ind

........ 4

6 K 2 + 4- 2 ( 2 + 3 ) 2 2 + -- = 0... 5

a luesofk [ thecaseofk = 0j ustifies

3 ) ] . T h e fi r st o f t he s e e u a ti o ns o f

H. Thesecond if f o rbre ityw eput

K 2 + 2 -i K k . .. .. .. .. .. .( 6 ,

+ 2 - ( + 3 )

ss i e l y K , K , K 0 , . . . & a mp c . a l l i n t er m s

d i ff e re n ti a l e u a ti o n ( 2 i s s at i sf i ed b y ( 3 )

K 4 a r bi t ra r y a n d th e o th e r co e ff i ci e nt s

sprocessforcompletingtheso lution

remar s: The indeterminatenessofK4isacircumstancethatadmitsof

.Itisone ofthearbitraryconstantsin

e uation. Itshowsthatw emaygi e

w e pl e as e e e n i f H = 0 a n d th e n

mpanyourarbitraryK4 w iththecorrespondingv a luesofK 6 K 8 & amp c. w esha llha easeriesw hich

atthatw illsatisfy thee uationw henthere

gforcew hate er andw hichthereforemay

yanynumber tothee pressiondetermined

enforce. Inthene tsectionw esha ll

actly sim ilartothis. Yetthisob ious

onof thiscircumstanceappearstoha e

hehasactuallypersuadedhimselftoadopt

n 3 L / 4r g i n L ap l ac e s .

a a i n La p la c e s .



s s_ u s e # p d



ES

ingthegenerale uationamongthe

/1

k ) - ( 2 c 2 + 6 ) K 2 + 4/ + 2

cei edthatthismustapplyw henk = 1

4 andthusapplyingthesamee uation

swhichoccursin thedenominatorofthe

6 .1 2 b m/ l ( 2 . 2 + 6 . 2 2 b m /l

22+ 3 . 2- 2. 2+ 3 . - inaninf initecontinuedf raction. A nduponthishefoundssome

aptedtodifferentsuppositionsofthe

te asathinguponw hichnoperson

ha eanydoubt thatthisoperationis

natthetimew henIf irstreadthis

nclusion andshowedmethatLaplace

+ 2/ 2 v anishesw henk isinf inite ly

annotbutbee ua ltothecontinuedf raction.

ethecase ifK 4hasanyotherv a luethanthat

K 2 + 2 / 2 c a nn o t th e n co n e r ge t o z e r o

a luesofk . B utunlessK 2 +2/ f2 is

isinfinitelygreat thesecondtermofthe

isinf inite ly smallincomparisonw iththe

mately

” 6 2 + 2

= 2 + 4 K 2

t.Nowthisis preciselythedegreeof

f t he c o ef f ic i en t s of x 2 , X 2 + 2 & a m p c . i n th e

2 . H en c e w h en x i s i nf i ni t el y n ea r ly e u a l

n d so a l so i s V ( 1 - X 2 d a /d , o r d a/ d O . N o w

o r w h e n x = 1 w e mu s t ha e d a /d O = 0

ofthedisturbancein thenorthernand

hecaseproposedforsolution byLaplace



s s_ u s e # p d



G E D ER RO R I N L A P LA C E S T H E O R Y 2 7

s e K - + S 2 / m u st c o n e r ge t o z e r o

ha ethev a luegi ento itbyLaplace* .

reeofcon ergenceobtainedby

ofK4 andverif y thatitsecuresda/d= 0

R . . .. . .. . .. . .. . .. . .. . .. . . 7 .

+ -. K 2 ;

f r R g i e s

3 + . .. .. .. .. .. .. . 8 .

2/ c+ 6 -R ) (

os c o n e r ge t o u ni t y ( 8 g i e s

w he n k i s g re at .. .. .. .. .. .. 9 .

nationofK 4byhiscontinuedf raction

nof theratiosby ta ingR + , =0 for

eofk andca lculating

n s of ( 8 w i th k - 1 k - 2 . . .s u bs t it u te d

t o th e s er i es ( 3 a d e gr e e of c o n e r ge n cy

sa m e as t h at o f t he e p a ns i on o f e X V + e - X V i n

s uc h th at d a/ d , d 2a /d 2 d a /d a , . .. & a m p c . a re a ll

a lueofx . Henceda/ dO , be inge ua lto

i s z e ro wh en x = 1 .

place sprocesssimplydetermines

nthatda/ dO = 0atthee uator. A ndthe

3 ) hasthere uisitecon ergency to

forthepoles. Laplace sresult isthereforethe

eproblemoffindingthe tidalmotion

ewholeearthcontinuouslyfrompoleto

rmotiontheseacouldha einv irtueofany

ot e ceptforcerta incrit ica ldepths ha e

ftheassumedtide-generatingforce.

ofsucha depththatsomeoneof

onsinwhichtheheightof thesurfaceat

leby theformulav cos2- w here r

mar sinreply Phil. Mag. Oct. 1875 re ferredspecially to



s s_ u s e # p d



ES

v somefunctionof the latitudeha ingthe

northandsouthlatitudes hasitsperiod

egeneratinginf luence it iseasily seen

f ferentia le uation( 2 gi esaninf inite ly

lywhenthedepthhasoneof these

bitrarysolutionsintroducedby Airyand

pplicabletoanoceanco eringthew hole

nto thesameerrorofimagining

onof thedif ferentia le uation( 2 , w ith

nts includesoscillationsdepending

f thesea asthefo llow ingpassage( Li . I .

L i n te g ra t io n d e l e u a ti o n ( 4 * d a ns l e c as g e ne r al o u i n n e s t

uneprofondeurv ariable surpasse les

maispourdeterminerlesoscil la t ionsde

asnecessairede l integrergenera lement i l

a ri lestcla irq ue lapartiedesoscil la t ions

primitifde lamer adubient6tdispara itre

ugenrequeleseau de lamereprou entdansleursmou emens ensorteq uesansF actionduso leil

raitdepuislongtempspar enueaunetat

e : F actiondecesdeu astresl enecarte

ssuff itdeconna itre lesoscilla t ionsq uien

didnotsufferhimself tobe ledintow rong

on andheseemstoha eentirely forgottenitw henhegoesdirecttotherightresult w ithoutnoteor

singular processreferredtoabo e.

A iry a f terha ing inthepassage

a llowedthesamemisconceptionto fatally

ngwiththe solution closeswitha

ntwhichissufficientto showthegroundlessnessofhisob ectiontoLaplace sresult andtheuntenability

. Thispassagehasnotonlythe

thearticle whichprecedesit butit

decidedad ance inthetheorybeyond

o f wh i ch e u a ti o n ( 2 o f o ur n u mb e ri n g ab o e i s a



s s_ u s e # p d




erdid orsuggested andforboth

o te i t . [ A i r y " T i de s a nd W a e s " a r t.

u s in g t he m or e c om p le t e v a l ue s o f a th a t we h a e

e d to f o rm t h e v a l ue s o f a ' , b a n d u w e f in d

eriesof termsmultipliedby the indeterminateK 4. WemaydetermineK4 sothat foragi env a lueof

i s t o sa y s o t ha t i n a g i e n l at i tu d e t h e

h-and-southmotion.We mightthereforesupposeaneast-and-westbarrier( followingaparallelof

nthesea andthe in estigationwouldstil l

eha eacompleteso lutionforaseawhich

osecourseiseastand west.

sprocessby thecontinuedf raction

eterminationofK 4thussuggestedby

hichA iry smethodfa ilsthroughnoncon ergence thatistosay thecase inwhichtheproposedeastand-westbarrierco incidesw iththee uator. F orasw eha e

pa ce s d e te r mi n at i on m a e s d a/ d O = 0 w h en 0 = 7

enorth-and-southmotionz eroatthe

ousf romsymmetry orasw eseef romthe

a p l ac e L i . I . a r t. 3 ; o r A i ry a r t s .( 8 5 ,

0

a-4 sin0cos0cos2. . . 10

entofthedisplacementofthe water

e s ee t h at L a pl a ce s s o lu t io n ( 3 ) , w i th K 4

ergentfora llv a luesofx & lt 1. Therefore

rgentfora llv a luesof0& lt ~ Tr. Hence

w i th t h e fo r mu l ae w hi c h he g i e s i n hi s

u at io ns ( 3 ) a nd ( 4 , ( 5 , a nd ( 1 0 o f~ ~ 5 a n d1

completeandcon ergentnumerica lso lution

e semi-diurnaltideinapolarbasin

uallydeepfromeitherpoleto ashore

titudeonthenearside ofthee uator.

eha eseen doesthesameforahemisphericalseaf rompoletoe uator. B utforaseae tendingf rom

cidingwithacircle oflatitudebeyond

ionalcomponentofthedisplacementofthe waterin



s s_ u s e # p d



ES

mofsolution(sti l l how e er w ithbut

mustbesought becauseLaplace s

5 a bo e , c ea si ng t o co n e rg e wh en x ( o r th e

0 increasesuptounity fa ilstopro ide

ationof0f romzerotoanyv a luee ceeding

emethodsuggestedbyA iry w hene tended

e differentiale uationwithitstwo

mpletelysol estheproblemoffindingthe

na lseaofe ua ldepthbetweencoasts

rallelsofl atitude.

ssolutionforthewholeearth

w ef indintheMecani ueC elestethenumerica l

y (butnotq uoted becauseof thesupposed

chtheywereobtained .Theyareof

est( w henwek now themtobecorrect ;

es Imaybepermittedtoquotethem

wor ingoutnumericallytheprocess

6abo e forthreedif ferentdepthsof thesea

1 / 6 1 -2 5 o f th e e ar t h s r a di u s. T h e v a l ue s o f e

hesedepths are10 2-5 1-25respecti e ly andLaplacef indsforthesolution[ ( 3 ) ~ 5 inthethree

0 , a = H { 1 - 00 0 0. x 2 + 2 0 -1 8 62 . x 4 + 1 0 -1 1 64 . x 6

1 0 - 7 4 5 81 . x 1 2

6-0-0687. x8

22-0-0001. 24 ;

{ 1 -0 00 0. x 2 + 6 -1 96 0. x 4 + 3 - 2 4 74 . x 6

0 9 19 . x ' l + 0 - 0 07 6 . x 1 2

H { 1 - 00 0 0. x 2 + 0 - 75 0 4. x 4 + 0 - 15 6 6. x 6

0 09 . x O } .

ngLaplace sf irstdif ferentia le uation[ theonef rom

o f ~ 5 a b o e b yp ut ti ng 1 - /u 2= x 2 ( L i . i . a rt 1 0 ] ,

1 U / 2- 2e ( 1 - E /2 2 a = - - 8H 1 _- i/ 2 ,

sumption

A 2 2+ & a mp c .

sacompleteso lutionw ithtw oarbitraryconstants tobereduced

dt io n t o ma e u = 0 a t o n e p ol e ( s a y w h en = + 1 .

atedin theprecedingfootnotesufficesforthis



s s_ u s e # p d




achcasew ef inda= 0 showingthatthere

o les. Puttingx = 1 w ef indinthethree

p t h 1/ 2 89 0 o f ra d iu s ,

1 /7 22 5 ,

1 / 6 1- 25 , ) .

firstcaseshowsthat thetideis" in erted

here islow w aterw henthedisturbingbody

highwaterw henit isrisingorsetting.

( thatistosay forpo larregions thesignis

orethetidesaredirectforthis asclearly for

c a us e i n e e r y ca s e th e f ir s t te r m is + H 2 .

uestion( depth1/ 2890 , asw eseef rom

e theva lueofa increasesf romzerotoa

andthendecreasestothenegati ev a luestated

edf rom0to1 andthe intermediateva lue

sroughly -95 orthecosineof18~ . Hence

esare in ertedinthew holezonebetw een

thandsouthlatitude whilethroughout

hof theselatitudesthetidesaredirect.

eforthesecondand thirdofthedepths

atin thesecasesthetidesare e erywheredirectandincreasecontinuouslyfrompolestoe uator.

onforthe e uatorialtideinthe

earev ery interestingasshow inghow much

hanH( thee uil ibriumheight . U pon

atforsti l lgreaterdepthstheva lueofa

minutionhasa limit namely thee uil ibriumvalue w hichitsoonappro imately reaches. Tof indw hat

" b i en t 6t ) t a e t h e ca s e of e = , o r d ep t h

raroughappro imationtoR ta e

mu la ( 9 ~ 8 w i th k = 3 . T hu s we h a e

icationsof( 8 withk = 2 andk = 1 we

= - 1 04 .

eroughly

04 . x 4 + 1 0 4. 0 6 7. x 6 + 1 0 4 .- 0 6 7. 0 18 5 8

4 + - 00 8 2. x 6 + - 0 00 07 1 x 8 ,

16



s s_ u s e # p d



ES

depthis aboutase entiethofthe

untofe uatoria lt idee ceedsthee uil ibrium

npercent.

secondofLaplace snumerica lformulae

1 5 w e ma y i nf e r th a t wh e n e is i n cr e as e d

o10 thev a lueofaforanyv a lueofx

yto+ o thensuddenlybecome-o

y f romthattil l ithastheva luegi enby

henehasav a luee ceedingbyhowe er

uew hichma esa= + oc theva lueofa

sofx isposit i e anddiminishesthrough0to

a luesasx isincreasedto1 thatistosay

withtwo v erysmallcirclesof latitude

recttidesw ithinthesecircles andvery

undtherestof theearth. A se isincreased

stcrit ica lva lue thenoda lcirclese pand

w hene= 10theyco incideappro imate ly

hlatitude.F romthegreatnessofthe

ace sresultforthiscasewemay j udgethat

habo e10withoutreachinga second

ichthecoeff iciento fx 4 a f terincreasingto

es-oo. It isprobablethatthenoda l

earerthee uatorthan18~ Northand

alueisreached.W heneis increased

iro fnoda lcirclescommenceatthetwo

rdsandgettingnearerto theformerpair

themsel esaregettingnearerandnearer

rearedirectt idesinthee uatoria l

nthezonesbetw eenthenodalparalle lso f

e anddirecttidesin thenorthand

Thisis thestateofthingsfor any

thesecondcrit ica lv a lue j ustconsidered

eneisincreasedthrough thisthird

pairo fnoda lcirclesgrow soutf romthe

n ertedtidesatthee uator directt ides

nodalcirclesof thefirstandsecondpair

nesbetw eenthesecondandthirdnodal

e andstill asine erycase directt ides

es a fourthcrit ica lva lueofe introduces

es andsoon.



s s_ u s e # p d




few hichw eha ej ustbeenconsideringareofcoursethosecorrespondingtodepthsforw hichf ree

raltypesdescribedaresymperiodicwiththe

hefreeoscillationswithoutdisturbingforce

es s ed b y t he f o rm u la ( 3 ) o f ~ 5 w i th K = 0

6 + K s 8 + & amp c.

& a m p c . a re t o b e fo u nd b y g i i n g an a r bi t ra r y v a l ue

determiningtheratiosR 1 R 2 & amp c. by

ofLaplace sformula

3 R + ) . .. ( 8 o f ~ ] ,

ofk , commencingwithav aluecorrespondingtothehighestratio to-beusedincalculatingcoefficients

dR = -a thene tapplicationof

oo w hichisthetestthattheva lueofe

espondsto adepthforwhichthe period

nsise actlyhalf theearth speriodof

eratiosR , R B , R isane ceedingly

b ectofpuremathematicsorarithmetic.

apide tinctionof theerrorresult ingf rom

anitstruev a lueforR + l inthef irst

( 8 . Supposingk tobeso largethat

a s ma l l fr a ct i on w e k n o w t ha t t hi s i s so m ew h at

ue o fR , a nd t ha t e/ k + 1 ( k + 4 i s st il l

eva lueofR + l. Hencew eseeatonce

e ta e 0 i n st e ad o f R + , . I f w e ta e

e fo r mu l a gi e s 0 f or R , a n d th e n ra p id c o n e r ge n ce t o t he t r ue v a l ue s o f B R - l R - 2 & a mp c . I f we t a e R + l

6 w e ge t R = o o R _ = 0 a nd t he n ra pi d

v a l ue s f or R â € ” , & a m p c . B u t i f w e ta e f o r

an 2 b y a c er t ai n v e r y s ma l l di f fe r en c e

e lessthan2 + byacorrespondingv ery

henforR , av a lue lessthan - bya



s s_ u s e # p d



F T H E TI D ES [ 2 6

ence andsoon. A nyva lueofR + l

eparticularv a lue lastindicated w ill

nough leadtothedesiredv a luesof the

three fourormoresuccessi eapplicationsof theformulare uiredtodissipatetheef fectsof the

andinstructi earithmeticale ercise

1 a n d so o n d ow n t o ] R a n d th e n by s u cc e ss i e

h e fo r mu l a to c a lc u la t e R2 R , . . . R - l R .

rigorous o fcoursethe init ia lv a lueof

ttheendof theprocess butif thecalculationhasbeenappro imate(say w itha lw aysthesamenumber

nedineachstep theva luefoundfor

l v a l u e b u t 2 â € ” , o r m o re a p pr o i m at e ly

k A + 2 A n d if w e c ho o se f o r RI a n y ot h er v a l ue

k + 2 '

dbyan infinitelyaccurateapplication

thenw or upbysuccessi ere erseapplicationsof theformula w ef indforR av a lueappro imatelye ua l

asnotwarnedusof this onthecontrary

yfollowed wouldleadussimply tocalculate

fa ct i on a n d th e n to c a lc u la t e K 6 K s & a m p c .

andK 4bysuccessi eapplicationsof the

lationof theformula

merica lquantity&gt 1. Ta eanyv a lueatrandomforr0

r . . . bysuccessi eapplicationsof theformula . F orlarger

riw il lbefoundmoreandmorenearlye ua ltothesmaller

.

dstor0by there ersedformula

e init ia lva lueof roaga in theresult( unlesstheca lculation

ate ine erystep w illbeappro imate ly thegreaterroot

pp r o i m at e ly e u a l to 1 / ri . T hu s f o r e a m p l e t a e t h e

5 - 8 2 84 2 7.

imationstothesmallerroot ta e



s s_ u s e # p d



GEDER R OR INLA PLA C ESTHEOR Y

This e ceptw ithinf inite ly rigorousarithmetic

rgeva luesofk notthetruerapidly

e co e ff i ci e nt s K 2 , K 2 + 2 & a m p c . b u t sl u gg i sh l y

orespondingtotheratioR = 2 - 4. B ut

y isa o ided andatthesametimethe

uchdiminished byusingforthe ratios

ndfortheminthesuccessi estepsinthe

dfractionforRi.

o fR 1 R 2 & amp c. consideredasfunctions

undamenta limportance. Someof theremar able

esentsha ebeenalreadynoticed(~ 15

atase( w hichisessentia llyposit i e in

ncreasedf rom0to+ oo eachof theratios

, , . . . increasesf romzero eachonemorerapidly

dngorder untilR Ibecomes+ oo and

andagaingoesonincreasingtill it

suddenly -o andsoon. B utbeforeRi

me R 2becomes+ oo -oo andagain

Thesameholdsforeach oftheother

ase increasescontinuously eachoneof the

o = 6 â € ” = 5 -8 28 4

= 6 - = 5. 8 28 4

' 2 = 6 - 1 - = 5 8 27 7

' F 3

= 6 5 80 ,

r 4 6 - - = 5 06 7

' 5 6 -r - 10 7 2

6 6 - = 2 02 9

6 r 7 6â € ” = 1 72 5

8

r 8 = 1 7 16 .

tephadbeenrigorous w eshouldha efoundr7 = r7 ? ' 6 =r6 andsoon Insteadofcomingbac ontheva lue0assumedforro w ef ind

e gr e at e r ro o t of t h e e u a ti o n



s s_ u s e # p d



ES

ge ceptw henitsva luereaches+ ooand

orderinwhichthe v aluesofthe

ghccis asub ectofgreatinterestand

rescarefule amination. Ihopetoreturn

ny remar thattheformula(8 forca lculating

Thatnotwoconsecuti eratioscanbesimultaneously

ncreasesf rom-ooto0 R iincreasesf rom0

b u t v e r y sl i gh t ly g r ea t er t h an e / i ( i + 3 ) , a n d

ea c he s o o w h en R i + , = 2 i + 3 ) .

i s & g t 2 ( + 3 a nd t he re fo re w he n Ri + & g t 1 R i

thatintheseriesofcoef ficients

...

e cu t i e c h an g es o f s ig n . F r o m ( 2 ( 3 ) i t

ntisless inabsolutev aluethanits

esign e ceptw henthepredecessoris

fficientprecedingit andoftwocoefficientsimmediatelyfollowingachangeof sign thesecondmay

utif so onlybyaverysmallproportionof

utthroughnearly thew holerangeofv a lues

hangeofsignf rom say K itoK i+1

l i n a b so l ut e v a l u e .( F o r i ll u st r at i on o f t hi s s ee

e f o r hi s c as e o f e = 1 0 f o r wh i ch h e g i e s

18 62 K , = 1 0- 11 64 K 8 = - 1 - 10 47

1 2 = - 74 5 81 K 1 4 = - 2 -1 9 75 . .. & a m p c .

n entionw hichformsthesub ectof

reate tension asIhopetoshow in

ha enothithertofoundanytrace

entia le uations butIcanscarce ly

nsomeformorotherit isnotk now nto

eoccupiedthemsel esw iththissub ect.

it iso fe ceedingv a lueandbeautyasa

hod.AstoLaplace sDynamicalTheory

ha emuchpleasure inconcludingw ith



s s_ u s e # p d




atementbyAirywhichI findinhis

" a rt . ( 1 1 7 .

romourthoughtsthedetailso f the in estigation w econsideritsgenera lplanandob ects w emusta llow

endidwor softhegreatestmathematicianofthe pastage.Toappreciatethis thereadermust

ldnessof thew riterw ho ha ingaclear

simperfectionofthemethodsof his

sothecouragedeliberately tota eupthe

amenta lly correct( how e eritm ightbe

terw ardsintroduced secondly the

gthemotionsof f luids thirdly the

gthemotionswhenthe fluidsco er

butcon e ; and fourthly thesagacity

anecessary toconsidertheearthasa

dthes il lo f correctly introducingthisconsideration. The lastpo inta lone inouropinion gi esagreater

eboastede planationofthelong

ndSaturn.



s s_ u s e # p d



S C IL L AT I O N S O F T H E F I R ST S P EC I ES I N

F THETIDES.

calMaga ine L. 1875 pp. 279-284.

ationsof theF irstSpecies aresimpleharmonicoscil la tions inw hichthesurfaceof thew aterisa lw aysa

ndthea iso f rotation. The" t ide-generating

ssuchthatthee uil ibriumtide-he ight

met adenotingaconstant( ca lledthe

sh-A ssociationT idalC ommittee sR eportfor

ctionof the latitude. ( be ingsupposedk now n

inga afunctionofthelatitudesuch

o s at i s t he a c tu a l ti d e- h ei g ht a t t im e t a n d f o r th e

deepe erywhere it istobeso l edby

ofthe differentiale uation

= 4 e .. . .. . .. . .. . .. . 1 ;

thelatitude andeandf areconstants

s

us

titssurface

oe uatoria lcentrifuga lforce be ing

f theearth srotation

sea supposedsmallincomparison

thanr/ 50.

O scillationsoftheF irstSpecies" isthe

ationa l andforita isabout1/ 14ofn



s s_ u s e # p d



O S C I L LA T IO N S O F T H E F I R ST S P EC I ES 2 4 9

. E enforthis andmoredecidedly forthe

andso larsemi-annua l(declinationa l and

goodappro imationtotheresultm ightbe

0. Laplacedoesnotenteronthe integrationof thee uation butcontentshimselfbypointingoutthat

f rict ionw ill w hen = 0 causethe

same asthee uilibriumtide-height

ar fortnightlytheactualheightmustbe

uilibriumheightifthereis enough

rtnightafreeoscillation toasmall

nt.Theresult ofanytide-generating

gperiodwouldob iouslybemoreand

greementw iththee uil ibriumtheory the

e itnotfortheearth srotation. B ut

otation a long-periodtidedoesnot

mentw iththee uil ibriumtide if thew ater

andthesolutionof thebeautiful" v orte

disw hatisa imedatbyA iry and

onof theprecedinge uationforthe

it isreducedtothecomparati e ly simple

ea - = 4 e .. . .. . .. . .. .. . . 2

e s ( E n cy c lo p ce d ia M e tr o po l it a na , a r t. ( 9 7 .

( A p pe n di t o U n i t ed S t at e s Co a st - Su r e y Re p or t 1 8 74 ,

ristol September2 1875. -Withoutthissimplif ication the

usceptibleofnearlyassimpleaso lutionasw ithit. A ssume

s ( K is -f2il

1 - - 2 d a

2 da = _ Mi ( i + 1 ( K i + l -K i -1 ;

K / 2

ecoeff icientsK i w eha ethee uationofcondition

1- K - _ 3 = 4e 0

.

analmoste uallysimplesolutionofLaplace sgeneral

w hichhasbeencommunicatedtotheB rit ishA ssociationat

d andw illbepublished[ inf ra p. 254 a lso intheNo ember

alMaga ine.



s s_ u s e # p d



ES

reesw ithA iry se uationofart. ( 97 ( w ith

edepthconstantasw enow suppose it , andw ith

u a ti o n ( 2 8 8 ; b u t is s i mp l er i n f or m p a rt l y

e snotationpforcos0 . F oreachof

intheactua lcaseof theearthunderthe

moon thefunctione isgi enby the

. .. . .. . .. . .. . .. 3 ) ,

uil ibriumvalueof thetide-he ightatthe

hisv a lueof (, f indsanintegralo f the

assuming

4 PA 4 + . .. + B i i + & a mp c .

cientssoasto satisfyit.B utthis

ngthetide-he ightatthee uatore ua l

t. Thecorrectassumptionfortheparticularproblemproposed( orforanycase inw hichO in o l es

L i s

B 4 4 + . .. + B j i i+ & a m p c .

mption

B , 2 + . .. + B i + & a mp c .. .. .. .. .. .. . 4 ,

ndincludesoscillationsinwhich the

s . W i t h i t we h a e

ea = i { ( i + 4 ( i + 1 B + 4

B + 2 -4eB i ,

o4e0. Thus forthecaseof

- 2 i = 0 i = 2 & a m p c .: 2. - 1 . B 2 = 0

4eHl. . .. . . .. . . .. . 5 ,

4 e B 2 = - 1 2 eH J

1 B + 4 - ( i + 2 ( i + 1 B i + - 4eB = 0 ... 6

a l ue s o f i e c e pt 0 a n d 2.



s s_ u s e # p d



O S C I L LA T IO N S O F T H E F I R ST S P EC I ES 2 5 1

5 gi esB =0 andwiththisthe

B = e B . .. .. .. .. .. .. .. 7 ;

u cc e ss i e l y i = 4 i = 6 i = 8 . .. a n d us e i n

sofound w ecanca lculatesuccessi e ly

B o 0 B 1 2 . . . ea c h in t e rm s o f B , ; a n d w e

( 2 w i th o n e ar b it r ar y c on s ta n t B 0 w hi c h

+ B o .F ( z , e . .. .. .. .. .. .. .. 8 ,

d e no t es t h e fu n ct i on o f u a nd e e p r es s ed b y t he

hecoef ficientsca lculatedforthecaseB = 0and

e t h e fu n ct i on s i mi l ar l y fo u nd b y t a i n g H= 0

iryhaspointedout [ " TidesandW a es "

referencetoacorrespondingq uestioninthe

tides maybeassignedsoastoma ethe

entmotionof thew aterz ero inagi en

ase( thatis thecaseofsymmetry round

w e ha e [ A i r y a rt . ( 9 5 , o r La pl ac e L i . I V .

o f wa t er = 4 / 1 t - da . .. 9 ;

henorthandsouthmotionzerow emust

.. . .. . . .. . . 10 ;

( , e

~ * * - 11 .

b y t hi s e u a ti o n fo r a ny g i e n v a l u e of u w e

eterminateproblemoffindingthemotion

entide-generatinginfluencewhen

w holeearth theseaco ersonlyane uatoria lbe ltbetw eentw oe ualcircularpo larislands.

isin aseriesessentiallycon ergent

caseof thepolarislandsvanishing. F or

6 i n t he o r de r i nd i ca t ed a b o e a n d so



s s_ u s e # p d




essi e ly f romsmallertogreaterv a luesof i

. . .. . .. . .. . . 1 2 ,

i + 1 ( 1 2 ,

reaterandgreaterv a luesof i

m o re n e ar l y th e g re a te r i s i. .. 1 ) ,

u e z e r o or o t he r w e gi e t o B o ( u n le s s we

luefoundbyLaplace smethodbelow and

he calculationwithinfiniteaccuracy .

theva lueofe theseriese pressingthesolution

a lueofA & lt 1. Thustheso lutionisthoroughly

edcaseoftwo e ualpolarislandsofany

eult imatecon ergence isshow nby ( 1 )

theseries . 2 . 4 . 22

1- 2

theseriesfora becomesinf inite lygreat and

ite lygreatva lueforda / da unlessit

ciselythe particularv alueofB osought.

fa ilstodeterminethisva lue. Thusthe

caseforw hichitw assought thecase

place andta enbyA iryandF erre las

estigation- thatis thecaseof thew hole

er. HereLaplace sbril l iantprocess re ferred

dingNumberofthe Philosophical

oura idmar e llously .

................... 14 .

2 4 ( i + 1 } . ... .( 1

ppliedtoanymoderatelygreate env a lue

eataccordingtothedegreeofappro imation

g N+ 2 = o c a lc u la t e Ni a n d th e n b y s u cc e ss i e



s s_ u s e # p d



O S C I L LA T IO N S O F T H E F I R ST S P EC I ES 2 5

. N6 N4successi e ly . E uations( 7 ,

. .. . . .. . . .. 16 ,

2H

B - 4= e + 2N 3 + 2 A7. 7

tionis

4U 6 A

6 N 6. N 8

in d 1 8

” . . .. N

, . . .a r e fu n ct i on s o f e de t er m in e d by ( 1 5 .



s s_ u s e # p d



2 8

NO F LA PLA CE SDI F ER ENTIA L

H E TI D ES .

calMaga ine L. 1875 pp. 388-402.

ceanas arotatingmassoffrictionlessincompressibleli uidco eringarotatingrigid spheroidtoa

tely smallinproportiontotheradius and

nsundertheinfluenceof periodicdisturbingforces withthelimitationthattherise andfallisnowhere

allfractionof thedepth thecondition

locityofe eryparto f the li uidisthe

andtheassumptionthatthedistance

he disturbedwater-surfaceisnowhere

ofthedepth.Thislast assumptionis

atedbyLaplace impliedin andisv irtua lly

as s um p ti o ns ( M e ca n i u e C el e st e L i r e I . No . 3 6

ofthewateris smallincomparisonwith

andthatthehori onta lmotionissensibly

ationof thew ater-surfaceabo e

ndDsin0thesouthw ardandeastw ardhori ontalcomponentdisplacementsof thewaterattimet andatthe

isTr -0( ornorth-polardistance0

e" e uationofcontinuity [ M ec. dl.

ir y " T i de s a nd W a e s ( E n cy c lo p ce d ia M e tr o po l it a na , a r t . ( 7 2 ] i s

r

. .. . .. . .. . .. . ) ,

) d yE . 1 b is

+ h = O . .. . .. . .. . .. 1 6



s s_ u s e # p d



G RA T IO N O F L A PL A CE S E Q U A T IO N

oof thedepthof theseatotheearth s

le uations[ Mec. Ce l. Li . I. No . 36 ( M ,

h-e

h - e ( 2 ,

d 2

h sradius ntheangularv e locityof its

gra ityatitssurface andethe" e uil ibriumtide-height att imet andco- la titudeandlongitude0

ThomsonandTa it sNaturalPhilosophy

atw hichthew aterw ouldstandabo ethemean

datrestre lati e ly totherotatingso lid

stifthedisturbing forcewerek ept

lityattimet.

atthegenera lintegrationof these

atdif f icult ies andheconf ineshimself to

se thatinw hichy isafunctionof latitude

ameina ll longitudes. Inthiscasethe

beeffectedby assuming

j

. . .. . .. . .. . .. . .. . .. . . 3 ) ,

fo rce issuchthat

r . .. . .. . .. . .. . .. . .. . .. 4 ,

r e fu n ct i on s o f th e l at i tu d e o f w hi c h E is g i e n

efoundby integrationof thee uations. With

b i s an d ( 2 g i e

H-0. . . .. . . .. . . .. . . .. 5 ,

= E

. d E - Er ' d O

( 6 ,

. .. . .

” s i

.....( 7 ,



s s_ u s e # p d



ES

b H f ro m ( 5 , b y ( 7 a nd ( 8 ,

)

4n 2 c os 2 0

0

ntiale uationofthetides[ Mecani ue

o3 , e u at io n ( 4 ; o r Ai ry " T id es a nd

p ce d ia M e tr o po l it a na a r t. ( 9 5 ] . I t i s a li n ea r

hesecondorder thecompleteintegration

t h en c e b y ( 8 , a a n d b i n t er m s of 0 w i th

bedeterminedso astofulfilproper

11-17 be low . It isessentia lly inthe

it be ingthatinw hichitcomesdirect

ngitin thein estigation.Itoriginally

ni ueC eleste mas edsomew hatby theaddit ion

intermw hichgi esitadif ferentform ,

betterorsimpler butthisasitw ere

ggeststhefollowingv erysubstantial

a nd ( s in 6 0 E = . .. .. .. .. 1 0 ;

n O d O

1

0 a 2 s in 2 0

pu t c os 0 = / L a n d fo r b re i t y

= f . . . .. . . .. . . .. . . 12 ,



s s_ u s e # p d



T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 5 7

b e co m e

8 intheprocessbyw hich( 9 w asfound

ngtheresult inge uationby4m( sinO) slf+2

1 U " 2 -do

2 S y 1 2d

f ' ) U J 2 -f L2 d̂ u

4m ( 1 - L2 4 ( . .. 1 4 .

e f irstthecaseof ( = 0( f reeoscil la tions , andassume

+ + K K p + . .. + K I i + & a m p c .. .. .. .. 1 5 .

f 2 + ( f 2 2 - o + . .

+ & a mp c .. .. 6 ,

tofintegration.Nowlet odenotea

hat

ra ll yw ( i = F ( i - 1 . .. 1 7

ctionof i. B ya idof thisnotationw emay

i . .. . .. . .. . .. . .. . .. ( 1 8 ;

eni= 0

9 ,

ra llnegati ev a luesof i.. . . .. . . .. . . 20 .

17



s s_ u s e # p d




f f iculty inde e loping( ~ 7below the

andw or ingoutapractica lsolutionof the

stinterestingandinstructi eresults

edtothecaseofanoceanofuniformdepth

n q = 0 o r y = c on s ta n t . T a i n g th i s

.... 26 ,

m e mb e r of ( 2 5 w e h a e

+ ( - 2i + - 4f ) K ^ J [ I 2_ 18 + 9 - i & l t / K i - + -i - _- O ( 2 7 .

uesof iupto-2thise uationisanidentity

a n d fo r i = - 1 i t b e co m es

. .. . ( 2 8 ,

ry . F o r i = 0 i t b ec o me s i n v i r t u e of ( 1 9

. .. . . .. . 29 ;

o( 2 0 ,

a f i K o = 0 . .. .. . .. .. . . 3 0 ;

tu e of ( 1 9 ,

s 2- aI f K 1 - a C= 0. .. .. . 3 1 .

, i= 4 etc.

K 2

K O

0

1 = 1 7- 2



s s_ u s e # p d




w e s up p os e d 1 = 0 a n d so m ad e f o r th e t im e

usub ect. Now suppose( ) tobeanygi en

tualproblemoftides ofanyspecies

nc t io n o f M/ o r o f a n d V / - U 2 , i f w e

ucedbythechange ofattractionofthe

f f igure . A properw ayof ta inginto

successi eappro imationsw illbe

me w ithoutlosinggenera lity Iassume

c I + ( I 2 + . .. + cD i i + & a mp c .... ... . 3 8 ,

( i 2 & a m p c . a re g i e n c on s ta n ts e i th e r fi n it e i n nu m be r

torendertheseriescon ergentforv a lues

sedineachparticularcase. Withthisfor

be r o f ( 1 4 b e co m es

.. . .. . .. . .. . 3 9 ,

w e ha e

2 s+ + l 2 ( 1- W 2 , ( W )

1 - 2 ( / 2 - 2 2 K + 1 = - = m - ( i - 2 . .. .. .. .. ( 4 0 .

accordingtothisformula mustbemade

h of t he p ar ti cu la r e u at io ns ( 2 9 , ( 3 0 , ( 3 1 ,

, ( 3 6 , ( 3 7 when re uired.

econditionswhichmaybefulfilled

fthetwoarbitraryconstantsC andK 0

estigatethecon ergencyof theseries( 16

orthecompleteso lution. F orthispurpose

g ( 2 5 a s t he c a se f o r wh i ch 4 = 0 i n to t h e

2 ) ( ) K / + 1

1 - _ 2 2 . ( f 2 _ -K , _ 3 ) - 4 m ( D hi - i- _2 } . . .. .. ( 4 1 .

antclassofcases o fw hichthef irst

mathematiciansisthatsosplendidlyand

placeinthe processdefendedandcontro ertedinthetwoprecedingNumbersof thisMaga ine terms

thise uationare forinf inite lygreat



s s_ u s e # p d




parable inmagnitudew ithtermsof thef irstmember

ginfinitelygreatof theorder2. These

ny ise itherconstantore pressedin( 3 4

70+ 7eya. R eser ingthemforconsideration

t h at e c e pt i n t ho s e sp e ci a l ca s es K i m u st

sof ifulf i l moreandmorenearly thegreater

+ = 0 . . .. . .. . .. . .. . .. . . ( 4 2 .

eandrigorousso lutionof thise uationin

e

+ " ' i ( - l i + ~ + + c.... 4 ) ,

m p e . de n ot e t he r o ot s o f th e e u a ti o n y = 0 a n d 1 1 ,

' , & a m p c . co ns ta nt s. H e nc e fo r gr ea t v a lu es o f i K i m u st

a lto ( 4 ) w ithsomeparticularv a luesfor

& a m p c . B u t f o r v e r y gr e at v a l ue s o f i al l t he t e rm s

n e le a di n g te r m o r [ b e ca u se o f t he e u a l ro o ts o f

e leadingpairo f terms v anishincomparisonw ith

Hencew emustha e forverygreat

, o r K i = [ ' + I ' ( - 1 ] i

o. .. 4 4 .

- r a n d so o n

ftherootsp p , & amp c.isgreaterthan

and( 16 arenecessarily con ergentfora ll

1 t o / = + 1 a n d th e y ar e d i e r ge n t fo r

hese lim itsunlesscondit ionspropertoma e

= 0 " ' = 0 a r e fu lf il le d. B u t i f on e or m or e of t he

mp c. islessthanunity andptheabso lutely leasto f

ionally theseries( 15 and( 16 arenecessarily con ergentfora llva luesof / f rom-pto+ p andtheyare

sof / beyondthese limitsunlessacondition

0isfulf i l led . When7y=0hasimaginary

asIaminformedbyhimselfandProfessorC ay ley has

enera lterms thiscriterionforthecon ergencyof theseriesin

fortheintegralof

- + x ( x ) = 0

totheR oya lSociety o fw hichcerta ine tractsha ebeen

ngsfor1870 1871 1872.



s s_ u s e # p d



T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 6

1 theabso lutemagnitudeofe itherof thepair

2+ 2 , andw iththisunderstandingthe

n ergencyanddi ergencyho ldsasfor

s distinctioninthecircumstancesof

inthetwocases o f transit ionthrougha

absolutev alueofapair ofimaginary

ereisnodiscontinuitywhenpu is

houghthecrit icalv a lueV / a2+ / 2 ; in

ferentialcoefficientsbecomeinfiniteand

asedcontinuouslyuptoandbeyondany

erpretationofthecircumstanceswhen

inf luencethesolutionisane ceedingly

owhichIhope toreturnina futurecommunication.Theremainderofthepresentarticlemustbe

0ha ingtw orea lroots eachless

ofy= 0 andputu= z + p. Then

sofz , thedif ferentia le uation( 14

+ ~ ( c + d ) + e + z . .. .. .. .. ( 4

e fdenoteconstants. Thecompleteso lutionof

ationmaybefoundbyassuming

z + H 2 + & amp c . } ( 4 6

K 2 + K z ' + & amp c.

, & a m p c . i n te r ms o f H 0 a r bi t ra r y b y

f og z , z l og z , z 2 l og z , & a m p c . to z e r o a nd

K 2 K , i n t er m s of K , a n d H , e a ch a r bi t ra r y

& a mp c . p re i o us l y fo u nd . T hi s s ho w s th e k i n d o f

ompletesolutionof thee acte uation

sentsw henthev a lueof / passesthrougha

dhow thisdiscontinuity isa ertedbyan

stantsofintegrationin therigorous

Ho= 0intheappro imateso lution( 46 .

uestion( ~ 9 o fassigningthetwo

asto fulfilanyproperphysicalconditionsofour problem.F irst towor outthegeneralsolution



s s_ u s e # p d



ES

u s e ( 4 0 , a n d c a lc u la t e K 1 K 2 & a m p c .

K oarbitrary . Thusw ef ind

Do X i 2 + X i ) + ( ) + . .. 47 ,

~ ) , X ( l X , X i 2 , & a mp c .a re nu mb er s ca lc ul at ed by th e

s m r 7 0 i Y Y 2 & a m p c . t o ha e h a d an y p ar t ic u la r

ne d t o th e m a n d & l t o I & g t 2 , . . . to d e no t e

forew ebeginthearithmetica lprocess

si g ne d t o 4& g t o l D , D 2 & a m p c . s o th a t we

L D 2 = n 2 L & a mp c .. .. .. .. .. 4 8 ,

a n d no n 1 n 2 & a m p c . g i e n n um e ri c al

o f theprocessofca lculationofKi f rom(40

X i L . .. . .. . .. . .. . .. . .. ( 4 9 ,

r e c a lc u la t ed n u mb e rs . T he n w e ha e b y ( 1 5

C + /( , ) . K o + X ( H . L

a o + a lp + a 2I 2 + & a m p c .

3 2 2 + & amp c.

X + 2 + & amp c.

C + B ( , t . K o+ x ( / . L

2 ao + I / f2 al 2 + 3 ( / f 2- c ia 3 + & a m p c .

f 2 + & a m p . / S+ ( -o + .

o + -+ f 23 2 + ( f 2X 2 - ) 3 + & a mp c ... 50 ,

constantsof integrationsoasto

renderingtheproblemdeterminate.

ortwotypicalcases:-first thesea

by twov erticalclif f s secondly by tw o

ualdeepeningfromeachtoa single

nintermediateparalleloflatitude.

nbeabelto fw aterbetw eenvertica l

des e itherbothinthesamehemisphere

rsouth.Theconditionsofthiscase

ndsouthmotionof thewaterateither



s s_ u s e # p d




flatitude andtheyareto befulfilled

b y pu tt in g

........... 52

a luesofF A ( thatistosay thesinesof

Ifeachof these islessinabso lutev a lue

0 e a ch o f t he s e ri e s in ( 5 0 a n d ( 5 1 i s

wholerangeofv aluesoffucorresponding

F V " t h e si n es o f t he t w o bo u nd i ng l a ti t ud e s

ati e forsouthlatitude ife itherorbothbe

b yu si ng ( 5 2 , i n( 5 0 ,

) . o + x ( ' ) L= ......... 3 ) ,

C+ 1 ( / ) . K o + X ( " ) . L = 0

/ - / ( ) . ) L LA

- as ( 0 ) a ( / . .. .. .

X ( ' ) - a ( " ) . X A ( u ) LI

- / ( ' ) . a( / )

C a nd K o ( 5 0 a nd ( 5 1 g i e 1 f_ - d

ofA throughtherangeof thesupposed

llow ingformulae[ w hichit iscon enientto

( 7 , ( 3 ) , ( 1 0 , ( 3 8 , and ( 48 abo e gi e hthe

4thesouthw arddisplacement and

eastw arddisplacementof thew aterattimet

ongitude r:

s/ - ' - f 2 2 df c os ( C t + sr

}

f / l2 da f2 l - ) s 2 . .. .. .. .. ( 5 5 ,

andm/ rtheratioofe uatoria lcentrifuga l

ione pressesthemotionofthe

duetoadisturbinginf luence o fw hichthe



s s_ u s e # p d



ES

sEcos( at+ sf , Ebe inge pressedby

o + n l + n 2 pY + & a mp c . . . .. .. 5 6 ,

a m p c . a re a n y gi e n n um b er s .

io n ( 5 4 g i e s e c ep t in a c er ta in c ri ti ca l

esently C = 0andK o= 0 andtherefore

eterminate ly thatthere isnomotion that

beany" f reeoscil la tion o f theassumed

( 3 ) * , e c e pt i n t he c r it i ca l c as e a ll u de d t o.

einwhichthe denominatorofthe

0 v a n is h es o r

nitev aluestoCandK ounlessLis z ero and

g i e s

( )

.. ./ / . ( 5 8 ,

* ( ) * *

oofC toKo butlea ingthemagnitude

all thefundamentalmodesof

posedz ona lsea isso l edbygi ingtos

2 & a m p c . a n d fo r e ac h v a l u e o f s tr e at i ng ( 5 7 a s

onforthedeterminationofor.After the

dSturmandLiou il le itmaybepro ed

uationcannotha e imaginary roots

nitenumberof realrootsmoreand

ntw henta eninorderofmagnitude

eto largerandlargerposit i e orf rom

largerandlargernegati e . Inthecase

n d ne g at i e r o ot s a re e u a l u n e u a l in a l l

s = 2 & a mp c . .

n cy o f t he s e ri e s in ( 5 0 a n d ( 5 1 i t i s

~ 9 thattherebenoroot rea lor

oseabsolutemagnitudeislessthan that

inesperfectly theconf igurationof theassumedmotion and

odis2w r/ a orits" speed a- .



s s_ u s e # p d




of thetwoq uantit iesA andp . B ut

ebraicsignsta enintoaccount there

weenpu and' u ( thatistosay w hen7

v a lueofponthedirectrangef romtp to

si f un ct ion sa , ) , / ( p , X ( , ) , a / , ( / u ,

pr oc es se s( 5 ) , ( 5 4 , ( 5 7 , ( 5 8 , a nd in th ef in al

5 0 , ( 5 5 , i s d i s c on t in u ou s . W h y so me o r a ll o f

iscontinuousinthis caseisob ious:

eroa longanypara lle lo f latitude lim its

hesideonw hichthedepthispositi e

sof7y= 0 separatestheproblemintotw o

ndthemotionsof thewateronthetwo

w ash. A nimaginary rooto f7= 0ha ing

betweent and ' " , o rarea lrooto f

ute lygreatero fp andj A , andof

w eenthem renderstheseriesfora( ip ,

m p c . i n as c en d in g p ow er s o f A di e r ge n t fo r t he p o rt i on o f o ur

sbeyond+ sin- R . Stilltheso lutionof

n b y ( 5 5 i n t er m s of s i f u nc t io n s a ( , u , , / ( t u , & a m p c . e a ch c o nt i nu o us t h ro u gh o ut t h e ra n ge b u t ca l cu l ab l e b y

wersof psetforthin ourpreceding

tof therangeoflatitudewhichli es

sin-1R.Themodeofdealingwith the

as toobtaincon enientformulaeforthe

( t &amp c. isaninterestingandimportant

toreturn. B e ing( ~ 9 atpresentlim ited

it isenoughtoremar thatinthiscase

t io n s a ( / u , 8 ( / u , & a m p c . a s er i es c o nt i nu o us l y

therangef rom// to / u maybefound

onsecuti erea lrootsof = 0 andlet

b e i n or d er o f a lg e br a ic m a gn i tu d e. L e t a be a n y

raically

a nd a& l t ( , A + p ) . .. .. .. .. .. .( 5 9 .

........ 60

d wo r i n g pr e ci s el y a s in ~ 5 b u t wi t h z i n s t ea d

mberof ( 15 andthepropercorresponding

amp c. w eobtainaso lutioninascendingpow ers

hisnecessarily con ergentthroughoutthe



s s_ u s e # p d



ES

degreeof con ergenceoftheseriesso

functions

( z + a , X ( z + a ,

( + a , X ( z + a ,

, thesameasthatof thegeometricalseries

c .

twoq uantit iesa-p p-a .

os ed c as e ( ~ 1 1 l et p , p p , p b e

o f ( 1 - / 2 y = 0 l e t p p b e e ac h b et w ee n

eposit i e forv a luesof / A betw eenp

determinetides andthef reeoscilla t ions

espondingtotheseintermediatev alues

ty a b e tw e en p a n d p , s u ch t h at p - a

hanthe lessof thetw odif ferencesa-p ,

a a n d so l e i n a sc e nd i ng p o we r s of z , a s i n

bethecoef f icientsofzi intheseriesthus

p , X 8 ) , ) i n fo rm ul ae c or re sp on di ng t o ( 5 0 , b ut

ondmembers sothatw eha e

C+ ( i ) o ( X i i L ... 6 1 .

a l ue s o f i a n d pu t

0 . .. . .. . . ( 6 2

q L =

b e e ac h i nf i ni t el y g re a t t h e v a l u es o f C a nd

e e u a ti o ns a n d us e d in ( 6 1 a n d ( 5 5 , g i e

eneratinginfluence

/ + n2 / 2 + & a mp c . .

entalfreeoscillationsof thesupposed

minedby f indingasoastoma e

. .. . .. . .. . .. . .. . 6 ) ,

ene pressedby ta ing

. .. . . .. . . .. . 64

of p - a= - p - a , w he n we m us t ta e p â € ” q = , o r an y

isbestinthiscase.



s s_ u s e # p d




. B y g i i n g mo d er a te l y gr e at v a l ue s t o p q ,

orousso lutionmaybesatisfactorilyappro imated

andnotvery laboriousarithmetic. The

9 ( 4 4 .

stigationstofindsolutionsforthe

orbothpo lesmustbereser edfora

l ehighly interestinge tensionsof

ocessreferredto in~ 9of thepresent

acesof the lasttw oNumbersof the

.

sby re uestk indly suppliedthefollowingnoteon

nhis discussionoftidaloscillationsis now

scorrect andthepaper( No. 26 on" ana llegederrorin

ac now ledgedtoha ef ina lly settledthecontro ersy

AiryandF errel.

r( No. 27 onthe" oscil la t ionsof thef irstspecies "

sof longperiod Ipo intedout( P roc. Roy . Soc.

p . 3 3 7 o r V o l . I o f c ol l ec t ed p a pe r s t h at L a pl a ce s a r gu m en t

amely thatf rict ionw asade uatetocausethesetidesto

umtheory . F o llow ingLordK el inIfoundnumerical

place smethod otherso lutionsha ebeenfoundby

~ 2 1 6 a n d by H o ug h ( s e e r e fe r en c es o n p . 2 1 . I t

ionsthatonan ocean-co eredplanetthee uilibrium

rror.

place sargumentasto f rict ion Ie a luated( Thomson

840 , ormyco llectedpapers V o l. I thee lasticy ielding

edf romobser ationof theoceanictidesof long

r ( B e i tr . z . G e o ph y si , V o l . 9 ( 1 9 0 7 , p . 4 1 , u s in g f ar

arri edataclosely sim ilarresult. Sincetherigidityof

sw ayagreesadmirablyw iththatfoundf romobser ationsw iththehori ontalpendulum w emay fee lconf identthatLaplace

ngtheseoceanictidesto conformtothee uilibrium

we erbee pla inedby f rict ion andatlength

gV o l . v ( 1 9 0 ) , p . 1 6 s h ow e d th a t la n d ba r ri e rs a s

nulthosemodesof f luidmotionw hich inthecaseof

et causesowideadi ergencef romthee uil ibriumlaw.

aplacewascorrectinfact asregardstheearth

g.

hesub ectw illbefoundinV o l. v iio f theEncy lopidiederMathematischenWissenschaften A rt. " B ew egungderHydrosphare .

n" thegeneralintegration ofLaplace stidale uation

stthanthetw oprecedingones since itssub ectisto

ed b y t he t w o me m oi r s of H o ug h ( s e e p . 2 1 w ho a l so

heeffectsof themutualattractionofthewater.



s s_ u s e # p d



[ 2 9

R .

A V E S I N F L O W I N G W A TE R .

ca l M ag a i n e x x i i . O c t ob e r 18 8 6 p p. 3 5 - 5 7 h a i n g

A of theB rit ishA ssociation B irm ingham

ebeautifulwa e-groupproducedbya.

hroughpre iously sti l lw ater butthe

islimitedtotwodimensionalmotion.

flowinginuniformregimethrough

hv erticalsides andbottomhori ontal

by trans erseridgesorho llow s orslopes

onta lbottomatdifferentle e ls. Included

sw emaysupposebarsabo ethebottom

tw eenthesides. Letthese ine ua lities

on A B , o f the length andletfdenote

of thebottom onthetw osidesof this

ebottombeyondA ishigherthanthe

i enataninf inite orv erygreat

petuallyflowingtowardsA withany

ocityu andf il lingupthecana ltoa

a.Itis re uiredtofindthe motion

throughA B , andbeyondB asdisturbed

weenA andB . Thisproblemisessentia lly

inasuf f icientlypractica lform theso lutionforthew a egroupproducedby theship w hichIhopetocommunicatetothePhilosophica l

nintheNo embernumber.



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I

onlyone solutionifweconfineitto

ca lcomponentof thew ater sv e locity is

mparisonw iththeve locityac uiredbya

heighte ua ltoha lf thedepth. Letb

v themeanhori ontalv e locityatv ery

and( toha ew todenotew a e-energy

w .. . .. . .. . .. . .. . .. . ( 1

ineticandpotentia l perunito f thecana l s

nto f itslength. Incasesinw hichthewater

eatdistancesf romB , w isz ero . B ut

isruf fled andthewaterf low s" steadily

andacorrugatedfreesurface asinthe

ofwaterflowinginamill-lead orHighland

uletontheeastsideofTrurnpingtonStreet

aceofPortland orIslayo erfa lls. The

esw hichweseeinthew a eofeachlitt le

w ould o fcourse e tendto inf inity if

long andthewaterabso lutely in iscid

d a si n gl e i ne u a li t y o r g ro u p of i n e u a li t ie s i n

mw ouldgi erisetocorrugationinthe

ssingthe ine ua lities moreandmore

hridgesandhollowsmoreand morenearly

ofthe canal thefartherwearefrom

tes. Obser ation w itha litt lecommon

lk ind showsthatatadistanceof tw o

omthelastofthe irregularitiesifthe

allincomparisonwiththewa e-length

en breadthsofthecanalif thebreadth

ththew a e- length thecondit ionof

straightridgesperpendiculartothe

dbefa irlyw ellappro imatedto e en

reasingle pro ectionorhollowinthe

tthesub ectof thepresentcommunicationissimpler asit isl im itedtotw o-dimensionalmotion and

s orridges orho llow s perpendicularto

us inourpresentcase w eseethatthe

rmityofthestandingwa esinthe

esisclose lyappro imatedtoatadistance

gthsf romthelasto f the ine ua lities.



s s_ u s e # p d



R

w ofi edv erticalsectionsof thecanalat

beyondA andbeyondB ; andp q the

seplanes.It willsimplifyconsiderations

SB atanode( orplacewherethedepth

a n de p th , a n d we t h er e fo r e ta e i t s o

saryforthefollowingk inematicaland

Thev olumesoffluidcrossingSA andSB inthesameor

l o r i n s ym b ol s

.. . . .. . . .. . . .. . . . (2 ,

lumeofwaterpassingperunit oftime.

e o r n eg a ti e o f t he w o r d o ne b y

w aterenteringacrossSA abo ethew or

lvo lumeof thew aterpassingaw ayacross

cessof theenergy potentia landk inetic

abo ethatofthe waterentering.

t a i n g th e v o l u me o f wa t er u n it y w e h a e

g b + ) ] . ... .. 3 ) .

( 3 .

tthef reesurfacezero w eha e

gb â € ” . . .. .. .. . ( 4 ;

ntitydependingonw a e-disturbance. Hence

w - w

+ = O . .. .. .. .. 5 .

2 = D ; a n d M = V D . .. .. . .. . .. . .. . 6 .

ndepth( intermediatebetweenaandb

ualtotheirarithmeticmean w henthe ir

parisonw ithe ither andV w illdenote

elocityofflow( intermediatebetweenu

mate lye ua ltotheirarithmeticmean w hen

comparisonwitheither .

5 g i e s

V 2



s s_ u s e # p d




ua lto f andif therew erenoberuf f lement

themeanle e lo f thew aterw ouldbethe

ea ingwateratgreatdistances onthe

thisisnotgenerally thecase andthere isa

e r i se o f l e e l g i e n b y th e f or m ul a

- ) - ( I . .. .. . 8 .

/ gD

nocorrugation( thatistosay o f

ormflow atgreatdistancesbeyondB .

0 a nd t he re fo re

/ i ) V \ ( 9 ;

byM2/ D2

) = ..........( 10 ,

D = . . .. .. . .. . .. . .. ( 1 1 .

betweenthesethreee uations

f It isclearthatthechangeof le e l

cientlygradualto ob iateanyofthe

dw henthisisthecase thee uationof

undf romy intermsof f fbe inga

ri ontalcoordinate x .

incomparisonw itha Disappro imatelyconstant[ muchmoreappro imate lye ua lto~ ( a+ b ] ,

nconstantproportiontof.

le thatV issmallincomparisonw ith

pr o pe r f ra c ti o n a n d y is a p pr o i m at e ly e u a l

asewhenV & lt V gDtheuppersurfaceof

ttomfalls andthewaterfallswhen

h en V & g t V I g D t h e wa t er s u rf a ce r i se s

r y pr o e c ti o n of t h e bo t to m a n d f al l s c on c a e o e r

andtheriseandfall ofthewaterareeach

rise andfallofthebottom sothat

18



s s_ u s e # p d



R

re le ationsof thebottom andisshallow er

bottom.

ecto f standingw a es( orcorrugationsof thesurface o f f rictionlessw aterf low ingo erahori ontal

ert icalsides Isha llnotatpresententer

ysisbywhichtheeffect ofagi enset

mitedspaceA B of thecana l slength in

onsinthew ateraf terpassingsuchine ua lities canbeca lculated pro idedtheslopesof the ine ua lities

ionsare e erywherev erysmallfractions

ongtocommunicateapaperto the

onthissub ectforpublication. Isha ll

hefo llow ingremar s: 1. A nysetof ine ua lit ieslargeorsmallmustingenera l

orrugationslargeorsmall butperfectly

arge shorto f the lim itthatw ouldproduce

u r a t ur e ( a c co r di n g to S t o e s s t h eo r y an o b tu s e

y trans erse lineof thew atersurface.

thewaterflowingawayfromthe

fectly smoothandhori onta l. Thisis

fo llow ingreasons: i Ifw aterisf low ingo eraplanebottomw ithinf initesimal

ualitywhichcouldproducesuchcorrugations

tomsoaseitherto doublethosepre iously

thesurfaceor toannulthem.

th( thatistosay the lengthf romcrestto

unctionofthe meandepthofthewater

orrugationsabo ethebottom andof

wingperunitof time.Thisfunctionis

nSto es stheoryof f initewa es. It is

t andisgi enby thew ell- now nformula

mal.

i t f ol l ow s t ha t a s i t is a l wa y s po s si b le t o

corrugationsbyproperlyad usted

itisalwayspossibletoannulthem.

cipleinthis modeofconsideringthe

erdisturbancetheremaybeinaperpetua lly

motionbecomesultimatelysteady all



s s_ u s e # p d




waydownstream.Thee planationof

e lopedinPartIII. tobepublishedin

edintheNo embernumberof the

a lhori ontalcomponentof f luidpressureon

tesinthebottom orbars w il lbefound

wor doneingeneratingstationary

iousapplicationtothewor donebywa ema ingintowingaboatthroughacana lw illbeconsidered.

onofthewa e-ma ingeffectwhenthe

maregeometrica llydef ined tow hich

red w illf ollow andIhopeto include in

tsinPartIII. tobepublishedinDecember

n i l lustratedbydraw ings o f thebeautiful

edbyaship propelleduniformlythrough

ca l M ag a i n e x x i r . N o e m be r 1 88 6 p p . 44 5 -4 5 2.

PartI. thesumofhori ontalpressures

bottom oronabar oronaseriesof

onsiderthehori ontalcomponentsofmomentumofdifferentportionsofthewaterin thefollowingmanner.

teady themomentumof thematterat

edv o lumeofspaceSremainsconstant

li eryofmomentumfromSbywater

egainofmomentumbywaterflowing

mustbee ua ltothetota lamountof

nthe waterwhichatanyinstantis

fthisforce beingthatoftheflow when

ingw atere ceedsthatof theentering

aceboundedbythebottom thefree

ndfourv ertica lplanes tw oof them ca lled

rtothestream andtw oof thempara lle lto

tancef romoneanother. Let$PB and

alinesonthetwotrans erseendsA andA oof the

b e i n g po i nt s o f th e s ur f ac e a n d B , B A p o i n ts o f t he

y



s s_ u s e # p d



R

ta lcomponentve locityatP . Therate

m( perunito f t imeunderstood f romSby

e ualto

1 ;

eryofmomentumfromSacrossA abo e

ossA 0ise ua lto

...( 2 .

thew aterbetweenA 0andA muste perience

ueinthedirectionf romA otowardsA made

ssuresonthe endsectionsA0andA

aterby f i edine ua lities if thereare

A . Hence ifX , X odenotethe integra lf luidpressuresonthe idea lplanesA A 0 andF thesumofhori onta l

l it iesonthef luid regardedasposit i e

tota lisf romA tow ardsA o ( 2 mustbe

. . .. . . 3 ) .

- ( X + u dy .. .. .. .. .. .. ( 4 .

P i s e u a l to g y + 1 ( q 2 - q 2 , b y t he

essureinsteadymotion( thepressureat

enasz ero , q andq denotingtheve locity

especti e ly .

2 -q 2 ] d y = I ( gD + q 2 D - q 2 dy.. . 5 .

D + q ) D + - v 2 d y. .. .. .. .. .. 6 ,

mponentv e locityatP .

ge pressionrelati elytoA0 gi e

mofhori ontalpressuresona ll ine ua litiesbetween

roblemof thef luidmotioninthecircumstancesissofarso l edastogi eD q , andu 2-v 2foreachof the



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I I

besofarontheup-streamsideof the

onofthe wateracrossitis sensibly

w ithve locityw hichw eshalldenoteby

0 ( 6 b ec om es

g D2 + U 0 D o . .. .. . .. . .. . .. . ( 7 .

a nd ( 4 ,

+ U 2 Do - q 2 ~ -D I- ( u 2 - d y. .. 8 .

locityatthef reesurface insteady

D o -D . . .. . .. .. . .. . .. . .. ( 9 ;

o B o f thebottombeingonthesamele e l

e lsbetweenthesurface-po ints$ 0

e co m es

U 0 ( D o -D - ( 2 - U 02 D

y. .. ( 1 0 ,

stantw hichmayha eanyva lue. It is

themeanhori ontalcomponentv e locity

reta e

.. . . . 11 :

ntit iesflow inginacrossA oandoutacross

m ot i on i s s te a dy w e h a e

. .. . . .. . . .. . 12 .

o f r o m ( 1 0 w e f in d

o- D 2 ~ ( + ( 2 + U 2 -U 2 dy .. 1 ) .

wek nowenoughaboutthemotion

eisrelatedtoother characteristicq uantities

9 , a nd i n it t a e

.. . .. . . .. . . .. . . .. 14 .

ta pointofthewater-surfacewherethe

e locity isrigorouslyorappro imately



s s_ u s e # p d



R

srigorouslyorappro imately thevertica l

. U s i ng n o w ( 1 4 i n ( 9 , w i th U D / D o

.. 15 ;

, g i e s

o + U 2 2

DU ] f V 2 + U 2 -U , 2 dy

2 . .. . .. . .. 1 6 .

eof le e l Do-D isbutsmall incomparison

e

( v 2 + U - 2 u dy .. .. .. 1 7 ,

matee ua lity . Go ingbac to( 16 , le t

ater-surfacethat

.. . .. . . . (18 ,

o becauseatacrestthef irstmember

andataho llow greater. Whenthe

mpleharmonic( thestream-lines

heposit ionof$ thuschosenw illbee actly

ndhollow.W henthemotionis

re a t u p t o St o e s s h i gh e st p o ss i bl e w a e

alessormoreroughappro imation

faw a e: it isa lw aysrigorouslydeterminate. F orbre ityw eshallca ll it thatistosayapoint

n o da l p oi n t. T h us w h en $ i s t a e n a s a no d al

messimplif iedto

D 2 ] + h i if v 2 dy .. 1 9 .

us.Init A whichisgi enrigorously

i m at e ly ( n o t ri g or o us l y e u a l to t h e v e r ti c al

: andifw esupposeDgi en Do isfoundby

b ic e u a ti o n in D o m o st e a si l y so l e d b y su c ce s si e

ngtotheprocessob iously indicatedby

uationappearsin( 15 . ( A saf irst

forDo inthesecondmemberandsoon.



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I I 2 7 9

19 forthecaseof infinitesimal

y ta eaatagreatenoughdistancef rom

rfaceinitsneighbourhoodbesensibly

themotionsimpleharmonic. The in estigationisfacil itatedbya lsota ing$ atanode asinthediagrams.

.. . .. . . . 20

reesurface thek now nsolutionforsimple

erofdepthDgi es

E -m D -y )

- s i n me

-D c o s m , . .. .. . 2 1 .

D+ mD

a s i n th e n od a l se c ti o n 3 P B ,

m t h m _ -m D. .. .. . 2 2 ;

2 mh 2. 2 )

. .. .. .. .4 D * . 2 4 .

9 w eseethatw henU approachesthe

D

eimportant e enthoughthecorrugationsata greatdistancedown-streamfromtheine ualities

ingconsiderationsofthiscase and

tobeconsiderablysmallerthanthe

ayneglectthef irsttermincomparisonwith

ingthatinfactq uantitiescomparablewith

dintheappro imation( 24 tothe

ndw eha e asourf ina lappro imate

2mD. .. . . .. . . .. . . .. 25 .



s s_ u s e # p d



R

derstandingthepermanentsteadinessofthemotionwhichweha enowbeenconsidering:toany

ergreat one ithertheup-streamordow nstreamsideof the ine ua lities if thew aterinthef initespace

hisstateofmotion andifw aterisadmitted

dawayontheother sideconformably.

ingandinstructi etoconsiderthe init ia tion

omanantecedentconditionof uniform

m. Suppose astheprimarycondit ion an

e le ationordepression toe istinthebottom

ththew ater sothattheflow of the

nformandinpara lle ll ines. If the ine ua lity isane le ationabo ethebottom oursupposit ionisthat

ece mo ingw iththew ater slipsa long

a litybeadepressioninthebottom the

ionmustbemadeofa plasticityofthe

f the ine ua lity carrieda long w hilethe

nebeforeandafter thisdepression.

ne ua lity isgradua llyorsuddenlybrought

eresult ingmotionof thew ater The

that offindingthemotionofwaterin a

ternalforce suchasthatofatow ing- rope

denlysetinmotionthroughit or

calif theboatwereabeamfilling the

anal sothatthemotionofthewater

sional.Ihopeina laterarticle( Part

sentseries to in estigatetheformation

dngw a esinthew a eof theobstacle

nfartherandfartherdown-streamfrom

nha ingbecomesensibly steady inits

comingsotogreater andgreaterdistances

letionofthegrowthoffresh wa es.

reamfromtheinitiatingirregularity

E uation( 15 showsthatw hetherthe

tion asinourf irstdiagram( f ig. 1 , o ra

a risingof le e lmusttra e lup-stream

ytothew aterw hichw ek now mustbe

sintermediatebetweenDoandthesmallerdepth

intheundisturbedstreamabo e. B ut

nit iatingirregularitymayha ebeen



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I I

l ingofane le ationup-streammustde e lop

locityofpropagationis asitw ere dif ferent

ope be ingV gD atthecommencement

ngfromthis throughV gDo , toV gD0asthe

o sothat asitw ere thebrow of the

up-streamo erta estheta lus t i l lthe

forourappro imation. The ine itable

w ater( ine itablew ithoutv iscidityof the

eactionpre entingthee cessi esteepness

-streaminamannerwhichitis difficult

therefore interestingtoseehow itmay

bysurface-action orbygi ingsomev iscosity

erestingtodothisby surface-action

beperfectly in iscid sothatourstanding

ybeperfectlyunimpaired.Andwemay

eringthef reesurfacea llo er( up-stream

haninf inite ly thinv iscouslyelasticfle ible

rans erse ly ( a f terthemannerof thesa il

by rigidmasslessbarswithendstra e ll ingup

desonthesides ofthecanal.Ifwe

eseendstoberesistedby forcesproportionaltothe irve locit ies andthemembranetoe ercise(posit i e

tractile tensiona lforce insimpleproportionto

eof itslengthineach infinitelysmall

hanica larrangementbywhichisrea li edthe



s s_ u s e # p d



R

fasurfacenormalpressurev arying

onentv elocityoftheotherwisefree

roportiontothis normalv elocitywhen

yma ingthev iscousforcessuf f iciently

theprogressof theriseof le e lup-stream

andperfectlya o idthebore . Wemay

fthe processionofstationarywa esdownstreamasslowasweplease.Theform ofthewater-surfaceo er

ua lit ies andtoanydistancef romthem

stream isnotultimatelyaffectedatall

n g a n d it b e co m es a s t im e ad a n ce s m or e

mathematicalsolutionforsteady

gi e w ithgraphicil lustrationsdraw n

mthesolution inPartIII.

ca l M ag a i n e x x I I . D e c em b er 1 8 86 p p . 51 7 -5 0 .

wemaynowconsidertheapplication

pedinitandinPartII. to thequestion

dweshallfindalmostsurprisinglya

nde planation 49~ yearsaf terdate o f

E perimenta lR esearchesintotheLaw s

lPhenomenathataccompanytheMotion

andha enotpre iouslybeenreducedinto

w nLawsof theR esistanceofF luids , "

ishsystemof" f ly -boat carrying

wandArdrossanCanalandbetween

ntheF orthandC lydeCanal a tspeeds

lesanhourtbyahorse orapa iro f

theban .Thepracticalmethodoriginated

itedhorse whosedutyitwastodrag

ed( Isupposeaw al ingspeed , ta ing

rawingtheboataf terhim andsodisco eringthatw henthespeede ceeded4/ gDtheresistancew as

ssell Es . M. A . F . R . S. E. R eadbeforetheR oyalSociety

18 7 andpublishedintheTransactionsin1840.

EnglishandA mericanrec oningofv e locity w hich

es1-609 3 k ilometresperhour or-44704metrepersecond.



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I II

Mr ScottRussell sdescriptionofthe

MHoustontoo ad antageforhisC ompany

ry isso interestingthatIq uote it ine tenso: C analna igationfurnishesatoncethemostinterestingil lustrationsof the interferenceof thew a e andmostimportant

cationofits principlestoanimpro ed

thediminishedanteriorsectionof

dbyraisingav esselwithasuddenimpulse

ressi ew a e thataverygreatimpro ementrecently introducedintocanaltransportow esitse istence.

n the iso latedfactw asdisco ered

wandArdrossanCanalof small

seinthe boatofW illiamHouston

prietorsof thew or s too f rightandranof f

anditw asthenobser ed toMrHouston s

hefoamingsternsurgew hichusedtode astate

andthevesse lw ascarriedonthroughwater

w itharesistancev erygreatlydim inished.

operce i ethemercantileva lueof this

withwhichhe wasconnected and

ducingonthatcana lvesse lsmo ingw ith

resulto f thisimpro ementw asso

epointo fv iew astobring f romthe

ersatahighv e locity a large increaseof

prietors.Thepassengersandluggage

oats aboutsi ty feetlongandsi feetw ide

ddrawnbyapair ofhorses.Theboat

behindthew a e andatagi ensigna l

o f thehorsesdrawnuponthetopof the

esw ithdiminishedresistance attherateof7

.

orsectionofdisplacementproducedby

ddenimpulseto thesummitofthe

snodoubtacorrectobser ationofanessentia l

n butit istheannulmentof " thefoaming

helowerspeeds usedtode astatetheban s

planationof thediminishedresistance.

atwhenthemotionissteady now a es

v o l . x i . ( 1 8 40 , p . 7 9.



s s_ u s e # p d



R

ttowedthroughacanal ataspeed

evelocityofaninfinitely longw a einthe

thew aterbe ingsupposedin iscid the

tbenilw hentheve locitye ceedsV / gD.

ouslyfortowageinani nfinitee panse

o eraplanebottom.

artII.for thewholehori ontalcomponent

torsuccessionof ine ua litiesonthe

culatethe resistanceonaboatof any

ppro idedw ek now theheightof the

owitsteadilyat itsownspeedinthe

reatdistancebehindit tobesensibly

hofthe canal accordingtotheprinciple

ofPartI. Theprinciple uponw hichthe

of f o rm u la ( 2 5 P a rt I I . m a yb e c al c ul a te d a re

inderof thepresentarticle andw illbe

PartIV .

f waterflowinginarectangular

w ithgeometrica lly specif iedine ua lities it

hemannerofF ourier to f irstsol ethe

hichtheprofileo f thebottomisacur e

esimally f romahori onta lplane.

a e O X a l on g t he m e an l e e l o f th e b ot t om

nofU themeanv elocityof thestream

sit i eupw ards. Let

. .. . . .. . . .. . . .. . . 1

bottom and

. . . .. . . .. . . .. . . .. . . . (2

reesurface f be ingheightabo eits

eve locitypotentia l u v theve locity

pressureatanypoint( x , y o f thew ater

e

. . . .. . . .. . . .. . . .. . . . 3 ) ,

+ ) . . .. . .. . .. . .. . . 4 .

uniformhori ontalv e locity isinf initesimal

are inf inite ly small. Hence( 4 gi es

- ( u -U ) . .. .. .. .. .. .. .. ( 5 .



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I II 2 8 5

f thee uationofcontinuity

rpresentcaseclearlyis

nm ( K e m Y+ K ' e -m y . .. .. .. .. .. . 6 ,

otionissteady K andK' areconstants.

) , gi es

m y + K ' e -m y . .. .. .. .. .. 7 ;

- K e -m Y . .. .. .. .. .. 8 .

o fyatthebottomandatthesurfaceare

especti e ly w ef indrespecti e ly forthe

elocityatthebottomandat thesurface

) , a nd m si nm ( K e mD - K ' / -m D .

ottom-stream-linesandsurface-stream-lines

theassumedforms( 1 and( 2 , w eclearly

H U . . . .. . .. . .. . .. . .. . .. ( 9 ,

D- K ' e - mc D = m s U . . . .. . .. . .. . 1 0 ;

H e - n D

. . .. . . .

hepressure isconstant andhence by ( 5 ,

c o ns t an t .. . .. . .. . .. . .. 1 2 :

( 7 , a nd ( 1 1 , w ef in d

( e + _ - - 2 H

1 ) ,

e 1 D - eD

rproblem forthecaseofthe bottoma.

ionof thebottomtobe

c 2c os 2m + K C c os 3 m + & a m p c . m A/ 7r .. . 1 4 ;



s s_ u s e # p d



R

face foundbysuperpositionofso lutions

a llowablebecausethemotionde iatesinf initely

formmotionthroughoutthew ater is

mA / 7r

e - .. . 1 5 .

( imD_ - imD

n( 14 bywhichthebottomis defined

w ell- now nsummationof itssecondmember

/ n A/ r. K ( c os m - K )

n A = 1 - 2 c os m + 2 +

ergentforallv aluesofK lessthanunity .

o fF ourier C auchy andPo isson the

inite ly litt le lessthanunityw illbemadethe

lso lutions. B y ( 14 w eseethat

.. . . .. . . .. 17 ;

uations( 16 w eseethat

2 _ _ .

.. . .. . .. . 1 8 .

K 2 . .. . ..

ittleshortof unitythefactorofd

8 isz ero fora llv a luesofx dif fering

r/ m(i be inganinteger ; andit isinf inite ly

7r/m. Hencew einferf rom( 17 and( 18

nalsectionofthebottompresentsa regular

nsanddepressionsabo eandbelow itsmean

nsbeingconf inedtoverysmallspacesonthe

o intsx = 0andx = 2i7r/m andtheprof ileareaofeachele ationbeingA . Thedepthsof thedepressions

e linthe intermediatespacesbetweenthe

ursee tremelysmallbecauseof thee ceeding

erw hicharethee le ations. F orour

on notonlymustA beinfinitelysmall

opeuptothesummit ofhmuste erywherebeaninfinitelysmallfractionofa radian andofcourse

loweringofthebottombetweenthe

s pl o tt e d in ~ 4 o f " D e e p W a t er S h ip W a e s " i n f r a.



s s_ u s e # p d




optionofameanbottom- le e lforourdatum

uced maybeleftoutof accountinour

ot aninfinitelysmallfractionofa

lho ld pro ideditsheightisverysmall

epthof thew atero erit . B utthe

dge wouldthennotbeits profile-areaA

er ofwhichtheamountwouldbefound

o erit f a renoughabo eittoha enow here

slope andfindingtheprofile-areaof

e itsow na eragele e lconsideredasthe

esee planationswesha llspea o faridge

n" irregularity or" obstacle " andca ll its

he" magnitudeof theridge" ; thisbe ing

themeasureof itspotency indisturbingthe

faridgeweha eaho llow A isnegati e

emay o fcourse ca llaho llow anegati e

n erges anddoesnotdependforits

glessthanunity sothatinitw emayta e

nity andw esha lldosoaccordingly .

singleridge remar thatifI bethe

....( 19 .

uriernow supposeIinf inite ly large w hich

l andput

. . . .. . . .. . . .. . . .. . 20 ;

5 b e co m es

s q (

- - .. . . .. . . .. 21 ;

U2/ g. . .. . . .. . . .. . . .. . . .. . . .. . . . 22 .

beshortened andforsomeinterpretations

q D = a w h en i t b ec o me s

/ D

.

Eâ € ” )



s s_ u s e # p d



R

1 or( 2 ) seemedratherintractable

uiredtoe a luate it f o rmanyandw idespreadenoughv a luesofx toshow theshapeof thesurfaceforany

D/b w ouldbev ery laborious. B utI had

uatingitfromtheperiodic solutionforan

uidistante ua lridges( 15 , w ho llyana logous

omcorrespondingsolutionsforcasesof

signallingthroughsubmarinecables to

and56ofmycollected Mathematical

nd tow ardsapply ingthismethodtoa.

bancedue toasingleridge Ihadfully

olutionfor thecaserepresentedbythe

g. 3, p. 295 , w henIfoundadirectandcomplete

single - ridgeprobleminaforme ceedinglycon enientforarithmetica lcomputation e ceptforthecase

o r f ro m z e r o t o a q u a r te r o r a ha l f of t h e de p th .

ppilygi esthesolutionforsmall v alues

a luesuptotw oorthreetimesthedepth by

ngseries andthusbetw eenthetw omethods

ysatisfactorysolutionof thew holeproblem.

ecur esandtheirre lationtotheproblem

allgi ethenew directso lutionof this

well- nownanalyticalmethodof

amplesaregi enintheEighteenthnote

i r on t h e Th e or y o f W a e s .

minatoro f (2 ) to theformof theproduct

uadraticfactors asfollows:-Let

- ( - e- . .. .. .. 2 4 .

o - w e ha e

l 2 ( 3 b

4 + & am p c . .. . 2 5 .

terthanD W isposit i e fora llrealv a lues

y po s it i e v a l u e l es s t ha n D W ( w hi c h

demieR oya lede ' InstitutdeF rance sa ansetrangers



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I II

mallv a luesofa2 isnegati e forlargev a lues

t le a st o n e po s it i e v a l ue o f 7 2 m a e s W z e r o.

hatonlyoneposit i ev a lueof -2doesso .

erosofW w henbisgreaterthanD and

thanD correspondtorea lnegati e

disob iousif fo rQ2w eput-02 w hich

0 ( 2 6 ;

0. . .. . . .. . . .D

zerosofW aregi enby therootsof the

nta le uation

Dthise uationhasa ll itsrootsrea l

f if th & amp c. q uadrants. Whenbislessthan

drantislost andinitssteadw eclearly

w h il e t he r o ot s i n th e t hi r d f i ft h & a m p c .

Le t 0 i 0 0 & a m p c . b e th e r oo t s of t h e fi r st

c . q u a d r an t s. A s t he f i rs t t er m o f e u a t i on ( 2 5 i s

1+ f. ) ( + ) .

a m p c . a re r e al p o si t i e n u me r ic s w h il e 0 12 i s r ea l

eaccordingasbisgreaterthanDorless

proca lo fW intopartia lf ractions w e

& a m p c . . .. .. .. .. 2 9 ;

-D/b cos0i

D / b- T co s i

0

' (

( 0



s s_ u s e # p d



R

b O i i s a s we h a e s e en i ma gi na ry ( i t s s u ar e

forthiscasetheformula( 3 0 maybecon enientlyw ritten

I - I ( t 2 a 1+ e -2 1

dingo- is

. ( e - e -u = O . .. .. .. .. .. .. . 3 2 ,

ne andonlyone rea lrootwhenD&gt b

lt b.

t i s ea s y to f i nd a s t he c a se m ay b e a o o f

t- u a dr a nt r o ot o f ( 2 7 b y a ri t hm e ti c al t r ia l a nd

e r o ot s 0 2 0 , & a m p c . m or e a nd m o re e a si l y

. I t i s to b e r em a r e d t h a t w h at e e r b e

serootsapproachmoreandmorenearly to

uadrantsinwhichtheylie:thus if

.. .. . .. . .. . .. . ( 3 3 ) ,

- D / b s in a i

. 9- /- / sc I ( 3 4 ;

1 7 r - a ] = D / b. c o s a. . .. . .. . .. . . 3 5 ;

orappro imationw heniisv ery large

= D / b. a s/ ta n a . . .. .. .. .. .. ( 3 6 ,

creasedto inf inity theva lueofa i

y toD/ b( i- ) 7r . Hencewheniis

ndmemberof (3 6 becomesappro imately

d th e e u a ti o n be c om e s

' 7 r _- D/ b. .. .. .. .. 3 7 ;

hthesmallerrootw henDislessthan3b and

Disgreaterthan3b isthere uiredva lue

) a n d m od i fy i ng i t b y ( 2 4 a n d ( 2 9 , w e

a / D

;



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T ER - II I 2 9 1

ll- now ne aluation( attributedbyCauchy

teintegralindicated

. . . .. .. . .. . .. . . 3 9 ;

b y ( 3 3 ) a nd ( 3 4 ,

( i - ]

D .. .. . .. . . 4 0 ,

d en o te a l l th e p os i ti e r o ot s o f ( 3 5 .

the ceedingrapiditywhenx isany

ndw ithv erycon enientrapidity forca lculationw henx ise enassmallasatenthofD. Whenx = 0 the

ally thesameorderasthato f1-e+ e2-&amp c.

f indthesumbyta ingasremainderha lf the

cluded.Thetruev alueofthesumis

v alueswhichweobtainbythis rulefor

s andthenforonetermmore. Whenit

sultwithconsiderableaccuracy alarge

ere uired anditw illnodoubtbe

ethodas indicatedabo e.

ef irsttermforthecaseD& gt b w hich

h e fo r m ( 3 9 b u t re a l in t h e fo r m ( 3 8

2. F orthiscasew eha e by thew ell now ndef inite integra l f irst Ibe lie e e a luatedbyC auchy

. . .. . .. . .. . .. 4 1 ;

e n by ( 3 2 a n d ( 3 1 * .

t i n as m uc h a s ( 3 8 h a s th e s am e

e a n d ne g at i e v a l ue s o f x , t h e e a l ua t io n s

a n d ( 4 1 a r e es s en t ia l ly d i sc o nt i nu o us a t x = 0

e - mustbesubstitutedforx inthe

rmulas. Ihope inPartIV . togi e

butwithorwithoutnumericalillustrations

re inthe integralsthe" principa lva lue o fC auchy is

ectstheinfiniteamplitudesinthe integrand which

ithf reev ibrations innaturesuchvery largeamplitudes

ictionalagencies andwhenthefrictionis slightthe

narrow confinedtothev erynearneighbourhoodofthe

ractua lcontributionisnegligible andthe" principa lva lue

fed.



s s_ u s e # p d



R

9 , w ith(41 foritsf irsttermandthe

ghoutw henx isnegati e isparticularly

uouse pressionforacur epassingcontinuously f romonetotheotherof thetw ocur es

--forlarge positi ev aluesofx

argenegati ev a luesofx . . . .. . . .. 42 .

D e e r y te r m of ( 3 9 i s r ea l a n d ( r e m em b er i ng t h at t h e si g n of x i s c ha n ge d wh e n x i s n eg a ti e w e

u a l fo r e u a l po s it i e a n d ne g at i e v a l u e s

symptotically toz eroasx becomesgreater

on. Ite pressesunambiguouslythe

uew henb&gt D o f theproblemofsteady

rmrectangularcanalinterruptedonly

tudeAacrossthebottom.This isthe

greaterthanthatac uiredbyabody in

ua ltoha lf thedepth.

ouni uenessofthesolutionwhen

ssthanthatac uiredbyabody infa lling

to h a lf t h e de p th ( b & l t D . F o r t hi s c as e

and( 41 e pressaparticularso lutionof the

througharectangularcanal when

nlyinterruptedbythesingleridge of

earlyha eaninf initenumberofso lutions

e instil lw aterinacanalo fdepthDw e

o f an y v e l o ci t y fr o m z e r o t o V / g D w hi c h i s

te ly longw a einw aterofdepthD. In

perimposeupontheso lution ( 3 9 ( 41 ,

trarymagnitude andarbitrarilychosen

eros w ithw a e- lengthsuchthatthe

agationisU , andthedirectionofmotion

essionofthewa etobeup-stream.

nstitutedconstitutesasetof freestationary

positionofthisupon thecaseofmotionrepresentedbyour symmetricalsolutionconstitutesthegeneral

ngle-ridgesteadymotion.To findthe

emustthusma etooursymmetrical



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I II 2 9

also lution put( 1 ) intothefo llowing

g ( D e â € ” D . .. .. .. .. 4 ) .

' . .

~ mayha eanyv alue( thatistosay

yw a esofanymagnitudeo eraplane

e m D = . . .. . .. . . ( 4 4 .

nowne uationtofindthev elocityU

ofperiodicw a esofw a e- length27r/ m in

satpresente uation( 44 istobe

cendentale uationfordeterminingthe

dingtoU agi env e locityofprogress and

en o n ly o n e re a l ro o t wh e n U & l t / g D b u t no

V / g D. P u tt i ng n o w in ( 4 ) U 2 = g b a n d co m pa r in g wi t h ( 3 2 , w e se e t ha t m D = o -r a n d go i ng b a c t o e u a ti o n

ha t

.. . .. . . .. . . .. 45 ;

raryconstants istheaddit ionw hichw e

t o g i e t h e ge n er a l so l ut i on f o r th e c as e b & l t D .

3 9 a n d ( 4 1 w e a cc o rd i ng l y ha e f o r

single-ridgesteady-motionproblem

V g D

C ~ W A/ D i N s n D + ~ i D

a n d

( ~ ^ A /D I s x A / ID o N I

.7 + E s C - D B N e D

D l b 2 %

.. . .. 4 6 ;

arbitraryconstants andAis theprofilesectionalareaoftheridge onthebottom.

ythissolution withanyv aluesofC

andstablethroughoutany f inite lengthof the

idge pro idedthewaterisintroduced

onsideredandta enawayattheother



s s_ u s e # p d



S O N W A T E R [ 2 9

tendstoinfinityinboth directions

utbegi eninthestateofmotioncorrespondingtothesolution( 46 themotionthroughoutany f inite

heridgewillcontinuefor aninfinite

6 . Thew ater ifgi enatrest m ightbe

otioninthefollowingmanner:- irst

eshaperepresentedbye uation( 46 , and

dtok eepite actly inthisshape so

twerein arectangulartubewithone

desplane andthefourthside( thebottom

placeof theridge. Ne tbymeansofa

allyin motioninthistube.To begin

he lidw ill inv irtueofgra ity benonuniform lessatthehighpartsandgreateratthe low parts. If

gi entothew aterby thepistonthe

ueof fluidmotion begreateratthehigh

arts. If thea eragev elocitybemade

surew illbeuniformo erthe lid w hichmay

the li uidisle f tmo ingsteadilyunder

ye uation( 46 asf reesurface. B utit

motionbeinggi entothef luidthroughout

ana loneachsideof theridge thatthe

oneachside oftheridgeconformableto

theparticularcaseof thisgenera lsolution correspondingto

/D . .. .. . .. . .. . 4 7 ,

4 6 t o

D w h en x i s p o si t i e

j E D w he n x i s ne ga ti e

sthemathematicalso lutionpromisedinthe.. . . .. . . . (48 ;

tion forthecaseofwater flowingfrom

o erthesingleridgeandtow ardstheside

mathematica lreali a tion forthecaseofa

cumstancesdescribedinPartI. abo e( ante

sthemathematicalso lutionpromisedinthe

Thedemonstrationthatthisis the

onforin iscidw aterf lowinginacana l



s s_ u s e # p d




dthee planationofhow anyotherstateof

a m pl e a s t ha t r ep r es e nt e d by ( 4 6 w i th a n y

butgi entothew aterthroughoutonlyaf inite

heridge settlesintothepermanent

edby ( 48 , mustbereser edforPartIV .

theJ anuarynumber.

ingdiagramrepresentsbytwocur es

46 fortheparticularv a lue2 456for

o r v e l o ci t y = ' 6 8 1 o f th e c ri t ic a l v e l o ci t y V / g yD .

ntsthesolution( 46 withC= 0 andC = 0.

esentsthepracticalsolution( 48 .These

ca lculationsofaperiodicso lution accordingtothef irsto f thetwomethodsindicatedabo e before Ihad

on( 3 9 byw hichthedesiredresult

edatw ithmuchlesslabour. Thefa intcur e

alculationfromtheperiodicsolution:

- 1 s h ow o n t he t w o si d es o f o ne r i dg e

romridge toridgeinthe periodicsolution

themiddleof thediagram. Thehea y

gtotheordinatesof thefaintcur ethe

ines foundby tria ltoasnearlyaspossible

andtodoubleontheotherside theordinates

wnearlyperfectwastheannulmenton

blingontheotheris illustratedbythe

ed( f ig. 3) , w hichhasbeendraw nby the

t imeslargercopy . How nearlyperfectthe

ngoughttobe atanyparticulardistance

easilycalculatedfromthesecondline of

ndw illbeactua lly ca lculatedforthecaseof these

ya lso forsomeothercasesfornumericali l lustrations w hichIhopetogi e inPartIV .

thepresentin estigationtotheef fecto fanine ua lityo fany

ea m i s gi e n b y V . E m a n A r ch i f i r Ma t em a ti ,

, B a n d 3 , N o . 2 1 9 0 6 .



s s_ u s e # p d



R

Y W A V E S O N T H E S U R A C E PR O D U C E D B Y

S O N T H E B O T T O M .

ca l M ag a i n e V o l . X X I I I . J a n ua r y 18 8 7 p p . 52 - 57 .

fsol ingthisproblemis bytheuseof

chweha ebeensow elltaughtbyF ourier

yofHeat andinthiswayit was

mulas1to15 ; theso lutionbeing( 15

.. . .. . . .1. . .. . . ( 1 ;

ancef romridgetoridge. Thus reproducing( 15 PartIII. withthenotationmodifiedtoshortenit in

mericalcomputation w eha e

(

e- i *

abo emeanle e lo f thew ater

o into eroneof the

lareaofoneof the

( 3 ) .

f P ar t I I I. ( 6 t o ( 1 8 o r

eane pressionforthesurface-effectofan

uidistantridgesonthebottom.W eshall

uccessionofridgesisfinite theresult

l lnotbeappro imatedtoby increasingthe



s s_ u s e # p d



A R Y W A V E S I N F L O W I N G W A T E R- I V

renceinthe effectofamillione uidistantridgesfromthatofa millionandonee uidistantridges

tionsonthesurfaceof thef luido erany

beasgreatas thedifferencebetweenthe

f athousandandone orbetweenthe

en: andtheabso luteef fecto f four orsi ,

bly thesameas ormaybegreaterthan or

ef fecto famillion inrespecttothecondit ion

pacebetweenthetwomiddle ridges.The

nsiderationofinfinityforourpresent case

w ith a f terthemannerofF ourier by

nitecana l" an" endless* canal " oracana l

t:acircular canalaswemayimagine

htbecur ed o fany form pro idedonly

cularornotcircular theradiusofcur ature

tcomparedwiththebreadthof thecanal.

necessarytoallowthe motionofthe

ecanal tobesonearlytwo-dimensional

imensionalmotioninastraight

pplicableto thewaterinthecur ed

gralnumbernofe uidistantridges

etabethedistancef romridgetoridge.

dditionofsolutionsofthe formula( 2

eef fect

)

e i - e -i

w ord" endless shouldincommonusage andespecia lly

odif ferentameaningf rom" inf inite . Thuse eryone

tbyan" endlesscord. A n" inf initecord means

ninfinitelylongcord-acordwhich hasnolimitto the

lusagein mathematicallanguage accordingto

e isca lleda" closedcur e " musthenceforthbeabso lutely

d toendlesstroubleinelectrical nomenclature according

nguage ane lectriccircuit issa idtobeclosedw hena

andtobeopenwhena currentcannotpassthroughit.

ta ll Englishw ritersone lectrica lsub ectsha ebeenguilty

whetheranyoneofthemwouldsaya roadroundapar

isclosed andisclosedw hene erygateonit isopen.



s s_ u s e # p d



R

ofdifferentv aluesofn e enorodd

tionsbothof mathematicalprinciples

dynamics butforthepresentI confine

I fo rw hich( 4 becomesidentica l

t i f M ( e i - e - i / e + e - i [ p r ac t ic a ll y c on s ta n t

saninteger thedenominatoro f (2 v anishes

othisinteger. Thisisthecase inw hich

thecanalis anintegralnumberof

o f f reew a esinw aterofdepthD. The

s andisinterestingbothinitselfandin

gproblemsin manybranchesofphysical

that w henthev a lueof

e- i

toany integerj , thechie f termof ( 2 is

andalltheothertermsarerelati e lyvery

ctisforcedstationaryw a esofw a elengtha/ . Thus ifw econsiderdifferentv e locit iesof f low

renearly totheve locityw hichma es

e-i aninteger themagnitudeof theforced

terand greaterforthesamemagnitude

nisstillperfectlydeterminate.Suppose

dgesmallerandsmaller sothatthew a eheightof thestationaryw a emayha eanymoderateva lue as

moreandmorenearly tothatwhichma es

e - i a n i nt e ge r t h e ma g ni t ud e o f th e r id g e mu s t

andinthe lim itmustbez ero . Thus

mayha estationaryw a esofanygi en

helim itingcase - thatinwhichtheve locity

e l o ci t y of a w a e o f wa e - le n gt h a / .

r th e c as e o f M( e i - e -i / e i + e - i a s f ar

ninteger thatistosay

- i j + . . .. .. .. .. .. .. . 5 ,

ora llv a luesof ilessthanj +1 the

sclearlynegati e w ithdecreasingabso lute

andfora llva luesof igreaterthanj it is



s s_ u s e # p d




sngv a luesf romi= j + 1to i= o . Thus

of thecoef f icientsofcosi rinthesuccessi etermsof theseriesf romthebeginningarenegati e w ith

esupto i= j ; andaf terthatposit i e

con ergingultimatelyaccordingtothe

hate= 2rD/ a w eseethatthecon ergence issluggishw hena thedistancef romridgetoridge( or

thecaseof anendlesscanalwithone

y large incomparisonw iththedepth butthat

epth ornotmorethanf i eortentimes

dingly interestingclassofcases , thecon ergence isv ery rapid.

how e er anotherso lutionstil lmore

morecon ergentindeedforthegreaterparto f

te erbetheratioofDtoa aso lution

entine erycasee ceptforv a luesofx

thedepth.Thecalculationforthese

ecessary togi etheshapeof thew ater-surface

ofthev erticalthroughtheridgesmall

th:for thispurpose andforthispurpose

2 indispensable . F orin estigatinga ll

tionthe newsolutionismuchmore

o l e s o n t he w h ol e v e r y mu c h le s s of

ndby summationfromthesolution

m gi e n i n Pa r t II I . ( 4 0 , ( 4 1 , a s

i dg e s be j + j ' + 1 a n d le t i t be

apeof thesurfacebetw eenthev ertica ls

+ 1andj + 2. Ta etheoriginof the

rticalthroughnumberj + 1 ridge andlet

heposit i esideof it . Theso lutionw illbe

lution( 40 PartIII. j so lutionsdiffering

i n g r es pe ct i e ly x + a x + 2 a . .. x + j a

j ' so lutionseachthesameas( 40 PartIII.

- x + 2 a . .. - x + j ' a s ub st it ut ed f or x . T h us

heef fectsof the j + j ' + 1single - ridges

1 -f + l f ix + ( 1 _f i ' ) f i i- l a.



s s_ u s e # p d



R

~ i

c o sa i

a

0 orthenumericbetw een

isfiesthee uation

n a i -D /b = 0

7 .

yof thef low

mridge toridge

onalareaofone ofthe

ontalcoordinatex , the

ethemeanle e lo f

herupstream or

es

gt D.Inthiscase asweha ealready

a , a c 2. .. a s a re a l l re a l a n d th e re f or e

achrea landlessthanunity . Hence inthiscase

s e ri e s o f w hi c h th e s um s a pp e ar i n ( 6 , a r e

d i f we t a e j = c m a nd j ' = o o ( 6 b ec om es

) . f i

ee pressionforSw hiche erof theridges

fx ; andtheva lueforx = a ise ua lto

Thew ater-disturbance isthereforee ua land

omridgetoridge andthesolution( 8 ,

e pressesw ithintheperiodtheheightof the

e e l n o t n ow a s i n ( 2 , t h e m ea n l e e l

buta le e latahe ight S. d / aabo e

b y i nt e gr a ti o n of ( 8 w e fi n d

. .. . .. . .. . .. ( 9 .



s s_ u s e # p d




rmingthesecondmemberof this

t ha t by ( 7 a bo e a nd ( 3 4 P ar t II I. w e ha e

' C os a C A l a. N i

/ D. s i n2 a i 1 - D /b )

a r t II I . ( 2 9 a n d ( 2 4 , w e fi n d

. .. . . .. . . .. . . 11 .

, ( 9 b ec om es

.. . .. . . .. . 12 .

e by Itheheightabo emeanle e l

f indf rom( 8 ,

.. . .. . . .. ( 1 ) .

thisand( 2 abo e twodifferente pressionsforthesameq uantity( with forsimplicity D= 1 , leads

abletheoremofpureanalysis

e -i

ai6- io+ e-0( a- ) 11

' 1 - e -~ a

enumeric

gt 1

etw eenzeroandv r/ 2

tion. . 15

t a n a- 1 /b = 0

7 r - a i

t i enumeric& lt a

asilyv erifiedbyta ing d c.cos --

memberofthe resultisob iously

- e - ) T h e se c on d m em be r m o di f ie d b y



s s_ u s e # p d



S P R O D U C E D B Y A S I N GL E I MP U L S E I N

P TH O R I N A D I S PE R SI V E M E D I U M .

ca l M ag a i n e V o l . x x I I i .M a rc h 1 88 7 p p . 25 2 -2 5 5 h a i n g

ya lSociety 3 rdF ebruary 1887 P roceedings

mplicity consideronly thecaseof tw odimensionalmotion.

nowofthemediumisthe relation

locityandthewa e- lengthofanendless

es. Theresulto fourwor w il lshow

progressofaz ero orma imum or

fav ary inggroupofw a esise ua lto

sofperiodicwa esofw a e- lengthe ua l

chmaybedef inedasthewa e- lengthin

particularpointloo edtoin thegroup

rallybe intermediatebetweenthe

nsideredtoits ne t-neighbourcorrespondingpointsontheprecedingandfollowingwa es .

e locityofpropagationcorresponding

TheF ourier-Cauchy-Poissonsynthesis

- t f m ] . . .. . .. . .. . . 1

dtime( x , t o faninf inite ly intense

me( 0 0 . Theprincipleof interference assetforthbyProf. Sto esandLordRay leighinthe ir

andw a e- e locity suggeststhefollow ing

l:Whenx - tf m isv ery large thepartso f the integra l( 1

ofasmallrange p- ato / + a v anish



s s_ u s e # p d



E S O N W A T ER [ 3 0

, be ingava lue ortheva lue o fm

] } = . . .. .. .. .. .. .. .. .. 2 ;

f ( A = V t . .. .. .. .. .. .. .. .. . 3 ) ,

f / L + , L f ' ( / * . .. .. .. .. .. .. .. .. .. . 4 ;

stheorem form-A v erysmall

/ x : -tf ) ] - t[ ~ f / ( ) + 2f ( ) ] I ( m - ) . .. 5 ;

,

= t { /2f ( A / + [ - Lf ( / - 2f ( A ] ( m -. 2 .. ( 6 .

1 -A d â € ” l â € ” . .. .. .. e

2f ( tA ] 2

1 , wef ind

A + o - 2

f ( p ]

einghere-ootooo becausethe

sso inf inite lygreatthat though + a the

are infinitely small amultipliedby it

2= . . . . . .. . . 9 .

- si n[ t L 2f / ( L ] 1 /2 co s[ t a 2f /( L + T r r

f ( / . ] 2 2 rr 2t f [ ( ) - 2f ( 4 ] ~ . .. .. .. .. 1 0 .

locityaccordingtoLordR ay leigh sgenera li a tionof

aresult. [ F orfurthere tensiononthe linesof thepresent

or e L ec t ur e s ' A p p. C p p . 52 8 -5 1 a n d pa p er o n ' D e ep - Se a

e d in f ra ~ ~ 8 0 s e . a l so H . L am b H y dr o dy n am i cs 3 r d e d.

. H a e l oc , P r o c . Ro y. S o c. A u g. 1 9 08 p p . 3 9 8 -4 0 a n d

E. v o l . x x i , J u ly 1 90 9.

at thisconditionofv erygreatdenominatorisnot

ssof tsuff icingby itse lf to j ustify the inf inite lim its: cf .

a .



s s_ u s e # p d



E S P RO D U C E D B Y A S I NG L E IM P U L S E 3 0 5

e-lengthandwa e- elocityforany

ar t h at b y ( 3 ) ,

- tf ( ) ] ,

toro f (10 ise ua lto / 2cos0 w here

] + I r .. . .. . .. . ( . . . .. . .. . ( 1 0

) ,

) ] } = 0

dO / d t = - l f p . .. .. .. .. 1 0 ) ,

sition.

f ir s t e a m pl e t a e d e ep - se a w a e s w e

.. . .. . . .. . . .. . 1

( 3 ) , a nd ( 1 0 t o

12 ,

t. . .. . . .. . . .. . . .. . ( 1 ) ,

gt27r

n = a os - ( 1 4 ;

2~ ^ , 4 4

son sresultforplacesw herex isv ery

hew a e- length27r// thatistosay

atgt2/ 4 isv ery large.

e s i n wa t er o f d ep t h D

- 2m .- .. .. . .. .. .. . 1 5 .

gh t i n a di s pe r si e m e di u m.

la ry g r a i t at i on a l wa e s

. .. .. .. .. .. .. ( 1 6 .

la ry w a e s

. .. . .. . .. . .. . .. . .. . .. . .. . .. . 1 7 .

20



s s_ u s e # p d



R

e s o f fl e u r e ru n ni n g al o ng a u n if o rm

. . .. . . .. . . .. . . .. 18 ,

e ura lrigidityandw themassperunito f

esha ebeenta enbyLordRayleigh

ra li a tionof thetheoryofgroup elocity andhehaspointedout inhis" StandingWa esin

ndonMathematica lSociety December1 ,

t an t p ec u li a ri t y of e a m pl e ( 4 i n r es p ec t t o

w hichgi esminimumw a e- e locity

ocitye ua ltow a e- e locity . The

entproblemforthiscase oranycase

minimumsorma imums orboth

ms o fw a e- e locity isparticularly

esnotpermit itsbeingincludedinthe

a n d ( 6 t h e de n om i na t or o f ( 1 0 i s i ma g in a ry

on f rom( 7 forwards gi esfor these

o f ( 1 0 , t he f ol lo wi ng :c os [ t 2f ( ) / ] + s i n [ t p U f ( a ] 1 9 .

t + 2f ( p ]

downforeachofthetwo last

a nd ( 6 ] .



s s_ u s e # p d



T A N D R EA R O F A F R E E PR O C E SS I O N O F

TER.

a n . 7 1 8 87 P h il . M ag . V o l . x x i i i.

11 - 1 20 .

perondif ferentlinesinPhil. Mag. O ct. 1904:

inaPaperinPhil. Mag. J an. 1907 ~~ 127-158:

ause inmyR . S. E. paperofF eb. 1 anditssuccessor

substanceof it w ithpromisede tensions isgi enin

e ea s il y r ea d f o rm . K . ( M e nt o ne M a rc h 3 0 1 9 04 .

S .

onw iththe InstitutionofMechanica lEngineers

A ug. 3, 1887.

Mech.Eng.1887 inPopularLecturesand

pp. 450-500 someof the il lustrationsbe ingomitted.



s s_ u s e # p d



[ 3 3

G A TI O N O F L A M I NA R M O T I O N T H RO U G H

V I N G IN V I S C ID L I Q U I D .

Re p or t 1 8 87 p p . 48 6 -4 9 5 P h il . M ag . V o l . x x I .

5 .

estigateturbulentmotionofwater

nes forapromisedcommunicationtoSection

tionatitsMeetinginManchester Iha e

lytowardsasolution( manytimestried

years o f theproblemtoconstruct by

ontoanincompressible in iscidf luid amedium

esoflaminarmotionasthe luminiferous

f l ight* .

dedona llsides andletu v , w

en t s a n d p th e p re s su r e at ( x , y z , t . W e

+ w ~ + 7 ~ ' x . . .. .. .. .. .. 2 ,

\ d dy d d ) ( 2

d dp

+ w ~ + ~ . .. .. .. .. . )

d y .. . .. . '

dp

d . .. ..

, ( 4 w ef in d t a i ng ( 1 i nt oa cc ou nt

d w 2 ( d d w d w dd w d ud

+ 2 - € ” + - ~ - + - --d d y \ d \ d d y d d dy d . ... ... .. ( 5 .

G e ra l d N a tu r e M a y9 1 8 89 P r oc . R oy . D ub . S oc . 1 89 9 a n d

o r i n Sc i en t if c P ap e rs 1 9 02 p p . 25 4 4 7 2 4 8 4. S e e al s o



s s_ u s e # p d



M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 0 9

o ne n ts u v w m ay h a e a n yv a l ue s

ac e s u b e c t on l y to ( 1 . H e nc e o n

w e h a e a s a p er f ec t ly c o mp r eh e ns i e

onatany instant

m + e o s ( n y+ f c os ( q z + ) . 6 ,

s ( m + e s il n( n y + f c O s ( q z + g .. . 7

os m) e o s ( n ym f s in ( q z + g ... 8 ,

f g ( e f a

q ) ' ( , q ) ' f e ) a r e an y th re e v e lo ci ti es s at is fy in g

g ( e f g ............ 9 ;

q ) + ' l ( m n q ) + q y m n q ) ............ 9 ;

on(orintegration fordifferentv a luesof

g . Th e su mm at io ns f or e f g m ay w it ho ut l os s of

f inedtotw ova lues: e= O , ande= 7r

= 0 a n d g = ~ 7 r . W e s h al l a dm i t la r ge v a l u e s

- l n- l q -1 undercerta inconditions[ ~ 4

( 1 2 , a nd ~ 1 5 be lo w , b ut o th er wi se w es ha ll s up po se

chof themtobeofsomemoderate or

linearmagnitude. Thisisanessentia lo f the

now proceed.

, x y a d en ot e sp ac e- a e ra ge s l in ea r s ur fa ce

telygreatspaces definedandillustrated

wo r e d ou t fr om ( 6 , ( 7 , ( 8 , a s f ol lo ws L

atlength orav erygreatmultipleof

n - , q - l m ay b e c on c er n ed : I1 ( L 7 c f g g

a .2 aA o ' , q ) c os ( n y+ f c os ( q z + g

.. .. . 1 0 ,

d d u = 2 a f ) c os ( ny + f . .. 11 ,

p r 0 0

- ) L j d dy d u = , ) .. ... ... ... 1 2 ,

e , ) ] 2 os ny + f cos2 ( q + g . . 1 ) ;



s s_ u s e # p d



R

sthat

e = 0 w e t a e 0 i n p l a c eo f

e = - 7 1 , ,

2 ( n . . . .. . .. . .. . .. . .. ( 1 4 ,

( r f g ( 0 , )

m n q )

f g

m n q ) ] c os ( n y+ f si n( n y f . .... ... . 5 ;

1 4 t h at

e= 0

e ad o f -

0andg= w ir

e= Tr 1 1

a n d g = 0 " 2 " 4

= T 7r n = O , f = - r , , , 1

nsfor( 15 .

e 2

( m n q ) ) * . . .. .. .. .. .. .. .. . 1 6 ,

rosofmandq, ana logoustothoseof ( 14 .

a eragingsforthepresent ta e

T hu s we f in d

t( e fg ( e f g ( e f g 2

8-2 { m a m n2 + q n ( m , q ) - + q y m , q )

7 .

ious.

ra lpropertyof thisk indofa eraging

.. . .. . . .. . . .. 18 ,

w hichisf inite forinf inite lygreatva lues

ntobe homogeneouslydistributed

iesthat thecentresofinertiaof all

idha ee ua lparalle lmotions if anymotions

herefore w eta eourreference linesOX ,



s s_ u s e # p d




e d r el a ti e l y to t h e ce n tr e s of i n er t ia o f t hr e e ( a n d

sof inertiao f largevo lumes inother

anslatorymotionofthefluid asawhole.

erylargea erageofmandof v andofw

mayremar , w ithreferencetoournotationof

s a sw es ee by ( 1 0 , ( 1 1 , ( 1 2 ,

= a n 0 q ) = a m n 0 = / 0 , q ) ... = ( m , , 0 . * ( 19 .

how e er encumberingourse l esw ith

nandnotationof~ 3 , w emayw rite asthe

null ityo f translationalmo ementinlarge

= a e w .. .. .. .. .. .. .. ( 2 0 ;

a eragethroughanygreatlengthofstraight

aofplaneorcur edsurface orthroughany

i ednotationofa erages homogeneousnessimplies

e v 2 = V 2 a e W 2= W 2.. . 2 1 ,

a ewu = CA a e u = AB . .. 22 ;

A B , C a r e si v e l oc i ti e s in d ep e nd e nt o f t he

whichthea eragesareta en. These

er inf inite ly shorto f imply ing though

ousness.

utionofmotionto beisotropic.

telymorethan isimpliedby the

ermsof thenotationof~ 8 w ithfurther

ew hatw esha llca llTHEA V ER A GEV ELOC ITY

R 2. . .. . .. . .. . .. .. . . ( 2 ) ,

. .. . . .. . . .. . . .. . . .. . . .. . . . (24 .

wpresentthemsel esastotransformationswhichthedistribution ofturbulentmotionwill

e li u idle f tto itse lfw ithanydistribution

initialdistributionbe homogeneous

esofspace e ceptacerta inlargef inite

hichthere isinit ia llye ithernomotion or

neousornot butnothomogeneouswith

roundingspace willthefluidwhich

uiremoreandmorenearlyastime



s s_ u s e # p d



R

ogeneousdistributionofmotionasthatof

tillultimatelythemotionishomogeneous

s coulditbethatthise ua li a tion

hsmallerand smallerspacesastime

words w ouldanygi endistribution homogeneousona largeenoughsca le becomemoreandmoref ine-gra ined

obablyyesforsomeinit ia ldistributions

obablyyesforv orte motiongi en

onelargeportion ofthefluid while

itia lmotiongi enintheshapeof

ho lt rings o fproportionssuitable for

andeachofo era lldiameterconsiderably

edistancefromnearestneighbours.

ghtheringsbeofv erydif ferentv o lumes

obablyyes if thediametersof therings

enotsmallincomparisonwithdistances

he indi idua lrings eachanendlessslender

ornearlyentangledamongoneanother.

: If the init ialdistributionbehomogeneousandceo lotropic w il l itbecomemoreandmorenearly isotropicastimead ances andult imatelyq uite isotropic P robably

tialdistribution whetherofcontinuous

orofseparatef initev orte rings.

metricalinitialdistributionof v orte

table .

nbehomogeneousandisotropic

andominrespecttodirection w ill it

yes. Iproceedto in estigateamathematica lformula deducible f romtheansw er w hichw illbeofuse

y ( 2 2 a nd ( 2 4 w e ha e

f or a ll v a l u es o f t. .. .. .. .. .. . ( 2 5 .

) w e fi nd

d ( u ) d( u ) dp dp

- z a u - -+ v - + w + v - + u --

d d dy . .. .. .. .. 2 6 .

e th e rh a ll A u g. 1 0 1 8 89 . S e e p. 2 0 2 su p ra .

nfactsuchfeo lotropyasthatof~ 20ismerely translationa l

rt ices. W. T .



s s_ u s e # p d



M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 1

d ( z v ) d ( u ) d p dp..

v + M s -+ V mu * * ( 2 7 .

c d y

sfor e eryrandom caseofmotion

lo w b e ca u se p o si t i e a n d ne g at i e v a l ue s o f

ua llyprobable andthereforetheva lueof the

isdoubledbyaddingto itselfw hat. it

, w w e su b st i tu t e -u - , - w w h ic h

a n d v e r if i ed b y l oo i n g at ( 5 , d o es n o t

etheinitial motiontoconsistof

, O , 0 s u pe r im p os e d on a h o mo g en e ou s

u 0 v o w ) ; s o t h a t we h a e

+ u 0 v = v o w = wo .. .. .. 2 8 ;

t o f in d s uc h a f un c ti o n f y t , t h at a t

componentsshallbe

w .. .. .. .. .. .. .. .. .. .. .. . 2 9 ,

uantit iesofeachofw hiche ery largeenough

o t ha t p ar t ic u la r ly f o r e a m pl e

z a v = x z a w .. .. .. .. .. .. .. .( 3 0 .

o r u v , w i n ( 2 w e fi n d

y t d u df

~ d y

3 1 .

. .. 3 1 .

fbothmembers.Thesecondtermofthe

ondtermof thesecondmemberdisappear eachinv irtueof ( 3 0 . Thef irstandlasttermsof the

r eachinv irtueof (18 a lone anda lso

0 . T h er e r em a in s

z a U d + v -+ w .... .. 3 2

econdmember[ by( 1 ]

d w) - + u + . ( 3 3 ) ;

- + U - .. .. .. .. . 3 3 ) ;



s s_ u s e # p d



R

pairoftermsofthe thus-modifiedsecond

1 8 f i nd

( u )

.. . .. . .. 3 4 .

t t hi s r es u lt i n o l e s b e si d es ( 1 , n o

( u v , w t h an ( 3 0 ; n o i so t ro p y n o

ct toy andonlyhomogeneousnessof

andz, w ithnomeantranslationa l

ncomponentofthemotioniswholly

, a n d s o f ar a s o ur e s ta b li s hm e nt o f ( 3 4

ofanymagnitude greatorsmallre lati e ly

oftheturbulent motion.Itisa fundamentalformulainthetheoryoftheturbulentmotionof water

ndIhadfoundit inendea ouringtotreat

erProf.J amesThomson stheoryofthe

niformR egimeinR i ersandotherOpen

a ouringtoad anceasteptow ardsthe law

armotionat differentdepths Iwas

seemingpossibilityofa lawofpropagation

inanelasticso lid w hichconstitutesthe

ommunication onthesupposition

tionU0 v 0 w 0isisotropic andthat

i de d by t he g re at es t v a lu e of f y t , i s in fi ni te ly

hesmallestva luesofm n q , inthe

, ( 7 , ( 8 f o r th e t ur b ul e nt m o ti o n.

eethat if theturbulenttmotionremained

casatthebeginning f y t w ouldremain

a luef y . Tof indw hetherthe

mainisotropict and if itdoesnot to

ow of itsde iationf romisotropy le tus

/ dt b y ( 2 a nd ( 3 ) , a s f ol lo ws :- i rs t b y m ul ti pl yi ng ( 3 1 b y v , a n d ( 3 ) b y u a nd a dd in g w e f in d

) d ( U v ) d f y t

W - f y t + ) } 3

y

15 1878.



s s_ u s e # p d




s andremar ingthatthefirst termofthe

by ( 3 0 , andthef irsttermof thesecond

w e fi nd w it h V 2 a s in ~ ~ 8 9 t o de no te t he

e locityof theturbulentmotion

) } - 2 t . .. .. .. .. .. .( 3 6 ,

( d ( v ) dp d ( 3 7

~ + w- + V d + y ( 3 7 .

dy

. 3 8 ,

w ouldbe if fw erezero . We f ind

d ( 3 9 ,

3 7 ,

d + u . . . .. . .. . .. . .. . .. . .( 4 0 .

de itherthesupposit ionof init ia l

motion oroftheinfinitesimalnessof

duceanduseboth suppositions.

ationof ( 3 9 , w enow useour

y t , di idedby thegreatestv a lueof

t el y s ma l l in c o mp a ri s on w i th m n q , w h ic h a s

e s

(

es

d d - d

+ U d -2 .. .. .. 4 2 .

i so tr op y w eh a e

0 A

d V * . 4 )

+ U o -+ W o - 7 V 02 V . 4 )

d dy



s s_ u s e # p d



R

ypartsforthelast twotermsofthe

us i ng ( 1 , w e fi n d

d u o dw d -

} ~ 0 = -x z a ( u + d - V o2

d d d y

2 o

) a n d ( 4 2 ,

d2 \ d d

o ( + + y V . -2 .. ( 4 4 .

u ri e r e p a ns i on ( 7 f o r v o w e f in d

f g c os ( a + e si n( n y + f c os ( q z + g

q ) 7 m2 + 2 + q 2 . .. .. .. .. 4 5 .

uf i e s & a m p c . d ro p pe d ,

- - m2 n2 + 2 .. . 4 6 * ,

n 2 + q 2

nSY M2+ q 2 / 2

-2 â € ” q . ... .. .

o = 8M 2+ 2 2 * 2 .( 4 7 .

a erageuniformityof theconstituentterms

omogeneousness( ~ ~ 7 8 9 , thesecond

u al t o â € ” 8 E E2 S / 2 a nd t he re fo re ( ~ 9

im ilarlyweseethatthesecondmemberof

2 R2 . He nc e f in al ly b y ( 4 4 ,

;

w i th ~ R 2 f o r V 2 o n a c co u nt o f i so t ro p y

} = - 2Rs dy t . 9

d( U V ) O - 9. t= 0 .. .. .. .. . 4 9 .

o

opy whichthise uationshowst is

o f thesmallnessofdf / dy and( 27 doesnot

ds i n v i r t u e of ( 3 0 . H e nc e ( 4 9 i s n ot

ues( v a luesfort= 0 o f thetw omembers

nitesimalde iationfrom2R2inthe

hana eragingthroughy -spacessosmallastoco erno

y t , butinf inite ly large inproportionton-1 isimplied.

W . T . A u g. 1 0 1 8 8 9 .



s s_ u s e # p d




member consideringthesmallnessof

fora llva luesof t unlesssofarasthe

referredtoattheendof~ 1 maybelost

rt icesv it ia ting( 27 ,

2R2 df y t ( 5 0 .

emberf romthise uation by ( 3 4 ,

51 .

yremar ableresultthatlaminardisturbance

othew ell- now nmodeofw a esof

uselasticsolid andthatthev elocity

/ , o rabout' 47of thea eragev elocityof

fluid. Thismightseemtogofar

ity tothevorte theoryof the luminiferous

edoubtfulpro isoattheendof~ 20.

tionof themediumbeastable

fv orte - ringsthesuggestedv it ia tion

annotoccur. F ore ample le titbesuch

w herethesmallw hiteandblac circles

g

" ' ; : - /

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ P 3 , H



s s_ u s e # p d



R

therings: thewhitewheretherotation

blac w here it isinthesamedirectionas

ofawatchplacedon thediagramfacing

aginef irsteachv orte - ringtobe ina

nedw ithinarigidrectangularbo of

edby thefinelinescrossingone

ughoutthediagram andtheother

per atanydistanceasunderw eli eto

olumeoftherotationallymo ing

utingtheringtobegi en there is

ape anddiametralmagnitude inwhich

erthatthemotionmaybesteady . Let

llspacew ithsuchrectangularbo esof

oneanotheroppositelyinthe manner

ulnowthe rigidityofthesidesof

ontinuesunchangedlysteady.B utis

igidpartit ionsaredoneaw aywith No

nthatit is. If it is laminarw a es such

uldbepropagatedthroughit andtheve locity

R/ V 2/ if thesidesof the idealbo es

dplanesof theringsares uare( w hich

e w 2 , a n d if t h e di s ta n ce b e tw e en t h e s u a re

stheproperratio tothesideof thes uare

a e U 2 = a e w2.

r e a m pl e p l an e w a e s o r l am i na r

erpendiculartothe undisturbedplanesof

nfigurationof thev orticesinthe

odofaharmonicstanding v ibration

sincy ( w hichismoreeasily il lustrateddiagrammatica lly thanaw a eorsuccessionofw a es , isi l lustratedin

hef luidoneachsideofy= 0. Theupper

entsthe stateofaffairswhent= 0

2 t o . B u t i t mu s t no t b e o e r l o o e d t h at

dependsontheunpro edassumptionthatthe

tisstable.

btful so farasIcanj udgeafter

tionfromtimetotimeduring theselast

theconfigurationrepresentedinfig.1 or

angement isstablewhentherigidity



s s_ u s e # p d




passim.

a n s an a e r ag e o f th e k i n d d e sc r ib e d in t h e fo o tn o te o n ( 4 6 ;

ebeinge panded

harebeingcontracted.



s s_ u s e # p d



R

singeach ringseparatelyisannulled

siblethat therigidityoftwo three

maybeannulledwithoutv itiatingthe

metricmotion butthatifit be

eofspace fora llthepartit ions the

able andtheringsshuf fle themsel es

re lati eposit ions w itha eragehomogeneousness l i e theult imatemoleculesofahomogeneousli uid.

erthesecondit ions the" v it ia tingrearrangement re ferredtoattheendof~ 20canbee pectednot

eperiodofaw a eorv ibration. To

meterofeachringtobev erysmallin

gedistancesf romneighbours sothatthe

srathertothe moleculesofagasthan

ouldnothe lpustoescapethev it ia ting

uldbeanalogoustothatin estigatedby

ek inetictheoryof thev iscosityofgases.

mt inconclusion thatthemostfa ourable

hepropagationof laminarw a esthrougha

iscidli uidistheScottishverdicto fnot



s s_ u s e # p d



2 1 )

O N O F V I S C O U S F L U I D B E T W E E N TW O

ca l M lf a ga i n e V o l . x x I . A u gu s t 1 8 87 p p . 18 8 -1 9 6

etheR oya lSocietyofEdinburgh J uly15 1887.

tionofthefirst ofthisseriesof

yof EdinburghinApril andits

phica lMaga ine inMayandJ une , the

steadymotionofav iscousfluidhas

ctfortheA damsPri eof theUni ersity

epresentcommunication( ~ 27-40

etwocasesspeciallyreferredtoby the

ouncement andpreparesthew ay forthe

simpleby apreliminarylayingdown

a t i on s ( 7 t o ( 1 2 b e lo w o f t he f u nd a me n ta l

av iscousf luidk eptmo ingbygra ity

boundariesinclinedtothehori onat

nw ithanymotionde iatinginf inite ly little

dymotionwhichwouldbetheuni ue

tionifthe v iscosityweresufficiently

almostcertainindeed thatanalysis

8and3 9w illdemonstratethatthesteady

iscosity how e ersmall andthatthe

ntedoutbySto esforty-fouryearsago

tigatede perimentally f i eorsi years

ds istobee pla inedby lim itso f

erandnarrowerthesmalleristhe

neof theboundingplanes para lle lto

al motion andO Yperpendicularto

1 8 87 p . 1 42 .

p. 166se . Thenumberingof thesectionsiscontinuous



s s_ u s e # p d



R E C TI L I NE A L MO T I O N O F V I S CO U S F L U I D 3 2

t o ( 1 a n d ( 2 , a n d in t h em s u pp o se

h in f in i te l y sm al l : ( 1 i s u nc h an g ed ( 2 , w i th U

b e co me

( d =

v C Y. .. 7 ,

) - ~ - d

dp

] V 2

d p

] = V 2 w- _ " ' ( 9 ;

sin I/ L. . .. . . .. . . .. . . .. . . .. . . .. 10 ,

denotes insteadofasbeforethepressuresimply

sI. y .

beafunctionofyandtdeterminedby

us ( 1 a nd ( 7 , ( 8 , ( 9 a re fo ur e u at io ns wh ic h

daryconditions determinethefour

v , w p i nt er ms of x , y z , t .

im inateuandw by ta ingd/ d ,

( 8 , ( 9 , a n d ad di ng . Th us w ef in d i n v i r t ue

. .. . .. . . ( 1 1 .

v /

uationsforthedeterminationofv andp.

m wefind

d2 _

c b 2- y2 ] = V 4 ... 12 ,

h w ithproperinit ia landboundarycondit ions determinestheoneun now n v . Whenv isthusfound

9 de ter mi ne p u an dw.

acticallyimportantcase ispresented

f theboundingplanestobek ept

thatis F and~ of ( 6 tobeperiodic

m pl e t a e

. . .. . .. . .. . .. . .. . .. . . 1 ) .

csolutionof( 4 is

b - y / o / 2 / /

/ - _ e -b -2 - co s ( o t - y 1 4 .

2~ 2 1- 2



s s_ u s e # p d



R E C TI L I NE A L MO T I O N O F V I S CO U S F L U I D 3 2 5

becomesreducedto

V 2

, ( 1 6 , ( 1 7 b ec ome

,

f l * d . ( 2 ) ,

2 4 ,

( 2 5 .

- d -. . .. . .. . .. . .. . .. . . 2 5 .

at e u at io ns ( 2 2 - 2 5 i mp ly ( 1 a nd

determinethefourquantit iesu v , w p.

noccasiona lly touse( 1 . We proceed

nof theproblembeforeus consisting

, w p s a ti s fy i ng ( 2 2 - 2 5 f o r a ll v a l ue s o f

n d th e f ol l ow i ng i n it i al a n d bo u nd a ry c o nd i ti o ns : wh e n t = 0 : u v , w t o b ea r bi t ra r y fu n ct i on s ( 2 6 ;

e ct on ly to ( 1 . .

= 0 f or y = 0 an da ll v a lu es of x , z , t .. . 2 7 .

w= 0 for y= b , ( )

a p a rt i cu l ar s o lu t io n u v , w p w h ic h

ditions( 26 , irrespecti e lyof the

7 , e c e pt a s f ol l ow s : = 0 w h en t = a n d = 0

andy= b. . .. . . .. . . .. )

a rt i cu l ar s o lu t io n u b b p s a ti s fy i ng t h e

darye uations: u= 0 b=0 to=O , w hent=0. . . .. . . .. . . . 29 ;

0 t u+ w = 0 w he ny = 0. .

olutionwillthenbe

v , w = t o+ w . .. .. .. .. .. . 3 1 .



s s_ u s e # p d



R

w r e ma r t h at i f p w er e z e r o t h e co m pl e te

dbe

y t ;

trialfo ratype-so lutionw ith/ notz ero

n -n ml t y + q z ] . .. .. .. .. .. .. .. .. . 3 2 ;

andtdenotes/ -1. Substituting

w e f in d

m/ t 2 + q 2 T .. .. .. .. .. .. 3 3 ) ;

n2+q + q 2-nZmt+ ( m2/ ) s2t2 ........... 3 4 .

, a nd ( 3 2 , w ef in d

mPt y + q z ]

_ M8 t 2 + q 2 . .. . .. . .. . .. . .

q 2 ( 3 5 ;

y+ q z ]

+ ] . .. .. .. .. 3 6 .

q 2 2

a n d pu t ti n g

( n -m n t y + q z ] . .. .. .. .. .. .. .. .. .( 3 7 ,

2/ m T

- 2 + q 2 q -

+ ( n -t + q ] - 2 + ( n _t 2+ q 2 2 3 8

e s W .

nd w w e fi nd u b y( 1 , a s fo ll ow s: n - m ft v + q w 3 9 .

. . .. . .. . .. . .. . .. . . 3 9 .

ddingtype-so lutionsfor+ b and+ n

w earri eatacompleterea ltypeso lutionwith forv , thefo llow ing- inw hichK denotesan

- t- t m 2 + n 2+ q ' 2 - t m p t+ ( m 2 /m ) 3 2 t 2 CO S

t 2 + q 2 s in [ m + ( n m ) q z ]

+ n mp t+ ( m 2/ ) 3 2 t ] C O S. .

2 si n- -7 -r [ m - n + mR t y + q q z ] J . .. 4 0 .

: s in I



s s_ u s e # p d



R E C TI L I NE A L MO T I O N O F V I S CO U S F L U I D 3 2 7

0

nn y ( m + q z ) . .. .. .. .. 4 1 ,

e ma e

42 ;

ersummationfora llv a luesof if rom1

rintegrationwithreferenceto mand

rminedva luesofK , a f terthemannerof

n y ar b it r ar i ly a s si g ne d v a l u e to V t = o f o r e e r y

o y = o , y = b. .. .. .. .. .. . 4 ) . Z = - o z = + C - C

ntegrationappliedto( 40 gi es

x , y z ; a nd th en by ( 3 8 , ( 3 7 , ( 3 9 w ef in d

tev aluesofwandu.

r bi t ra r y in i ti a l v a l u e w 0 t o t he z c o mp o ne n t of v e l oc i ty f o r e e r y v a l ue o f x , y z , a d d to t h e

, w h ic h w e ha e n o wf o un d a p a rt i cu l ar s o lu t io n

) f ul fi ll in g th e fo ll ow in g co nd it io ns : = 0 f or a ll v a l u es o f t x , y z

0 a nd a ll v a l u es o f x , y z

5 a n d ( 1 , b y r e ma r i n g th a t v ' = 0

p = O , a nd th er ef or e( 2 ) a nd ( 2 5 b ec om e

W ' . .. .. .. .. .. . ( 4 6 .

us ta swe sol e d( 2 1 , by ( 3 2 , ( 3 3 ) , ( 3 4 ; an dt hen

osatisfythe arbitraryinitialcondition

, ( 4 1 , ( 4 2 , w e ac hi e e t he d et er mi na ti on

w edeterminethecorrespondingu , ipsofacto

y puttingtogetherourtw osolutions w e

, w = w + w . .. .. .. .. .. . 4 7



s s_ u s e # p d



R

w ithout( 27 , inansw ertothef irstre uisit ion

to f i nd u b [ 0 i n a ns w er t o t he s e co n d

.

f irstf indingarea l( simpleharmonic

, ( 2 2 , ( 2 ) , ( 2 5 , f ul fi ll in g th e co nd it io n

n wt

ncotw heny=

n co t

snt ( 4 8 ,

s in c ot w he n y= b

ot

D E F , X 3 , 3 , S ) E , a re tw el e ar bi tr ar y

) . T h en b y t a i n g j d w f o o f e a c h of t h es e

rier w eso l etheproblemofdetermining

ughoutthef luid bygi ingtoe ery

imatelyplaneboundariesaninfinitesimal

hofthethree componentsisanarbitrary

t . L as t ly b y t a i n g th e se f u nc t io n s ea c h = 0

andeache ua ltominusthev a lueof

y po i nt o f e ac h b ou n da r y w e f in d t he u b t o o f

o fourproblemof~ 3 2isthencompletedby

Todoa llthisisamereroutineaf teranimaginary

dasfollows.

a s su me

q z ) J

) { H ey / m 2+ q 2 + K e -y / m 2+ q 2

[ y V ( W ' 2+ q 2 dye V ( , 2 + q 2 [ L f y M ( ]

d y/ .2 + q 2 [ f ( y + y M ( ] ] } . .. 49 ,

M a r e a r bi t ra r y co n st a nt s a nd f F a n y tw o

d - ( m 2+ q 2 . .. .. .. .. 5 0 .



s s_ u s e # p d



R E C T1 L IN E AL M O T I O N O F V I S CO U S F L U I D 3 2 9

ut 3 1/ LU = y andm2+ q 2+ /= ........( 51 ,

( X + V r y - . .. .. . .. . .. . .. . .. . . ( 5 2 ;

sc e nd i ng p o we r s of ( X + t y y , g i e s t wo

ichwemaycon eniently ta eforourf

= - 2( x + y y 3 7 -4 ( X + t y 6 7 -6 ( + v y 9 + & a mp

. 2 + 6. 5. . 2 9. 8. 6. 5. . 2

( X y y 7 7 6 ( X + t y) l + & a mp c .

+ 7 . 6. 4 . 1 0 .9 . 7. 6 .4 . . . .. . .. . . 5 ) .

sentia lly con ergentfora llv a luesofy .

easo lutioncontinuousf romy= 0toy= b

nstantswecangi eanyprescribed

/ d y f o r y = 0 a nd y = b . T hi s d on e f i nd p

; andthenintegrate( 25 forw inan

eriesofascendingpow ersofX + tyy w hich

butneednotbew rittendownatpresent

ll o ws : w = 0 t W t + m + q z ) . . .. . .. . .. . .. . .. . .. . ( 5 4 ;

+ K 2 ( X + t yy + L ( X + t yY

+ Pey ( n 2+ q 2 + Q e -y m2+ q 2 " ( 55

wof reshconstants duetothe integration

gi etoW anyprescribedva luesfor

y b y ( 1 , w i t h( 4 9 , w e h a e

m + q z ) I

- . â € ” ( 5 6 .

q . . . . .. . .. . .. . .. . 5 6 .

s an ts H K , L M P Q , c le ar ly a ll ow

e d v a l ue s t o ea c h of 6 V i W , f o r y= O

ompletionof therea li edproblemw ith

tions asdescribedin~ 3 7 becomesa



s s_ u s e # p d



R

e ( u v , w s o lu t io n o f ~ 3 4 c o m e s

asymptotica llyastimead ances asw e

3 4 , a nd ( 3 8 . H enc et he ( u b b O ) o f ~ 3 7 whi ch

att= 0 comesasymptotically toz ero

We concludethatthesteadymotion

W I N G D O W N A N I N CL I NE D P LA N E B E D .

ca l M ag a i n e V o l . x x i . S e pt e mb e r 1 8 87 p p . 27 2 -2 7 8.

ndofthe twocasesreferredtoin

ecaseofwateronaninclinedplane

d p ar a ll e l pl a ne c o e r ( i c e f o r e a mp l e , .

directionsandgra itye erywhere

e asasub-case the icyco ermo ing

w ithit w hichisparticularly interesting .

ngentia lforceattheuppersurface it is

hesamecaseasthatofabroadopenri er

perfectlysmoothinclinedplanebed. It

eptw henthemotionissteadily laminar the

surface isk eptrigorouslyplane butnot

cf . LordRay leigh " O ntheQuestionof theStabil ityo f the

P h il . M ag . x x x i . 1 8 92 p p . 59 - 70 : S ci e nt i fi c P ap e rs i . p . 5 82

forcedmotiondeterminedin~ 40 thef luidiscapableofa

hofwhichthev elocityattheboundaryisnull. Inthe

, w ( o r wh at i s th e sa me W , V a nd d V / dy i n ~ 3 9 , a t ea ch

y theratiosof thesi constantsare in o l ed toaperiod

oducingforeachf reeperiodnormaltypesofmotionw hosesca le

ined: imaginaryva luesof thesef reeperiodsmightin o l e

mis appliedbyLordRayleighhimselftothe argument

owing. Thee perimenta lin estigationofO sborneR eyno lds

ppeartoshow howe erthatw ithincerta inlimitso f theve locity

w ispractica lly stable . O nesuggestion mentionedbyLord

that asthere isnocontinuoustransit ionfromsteadymotion

motionofperfectf luidw ithnov iscosity theactua lmotion

stymay in o l e instabil it iesinav ery thinlayera long

. Inanycase the in estigationsinthete tw ouldperhaps .

nggeneralremar s stillretainanapplicationas determininghowfarsteadylaminarmotion ifsomehowestablished issusceptibleto

foutside forces.

LordR ay le ighnow referstoapaperbyProf . W. MC . O rr

v ii. No . 3, 1907 e tendinghisow npre iouscriticism

w ithw hichheisdisposedtoagree. Inthatpaper how e er

pp. 72 74 99 w hichareheldtoma eitprobablethatthe

eawaye ponentia lly andthattheforcedoscil lat iondeterminedinthete tistheactualso lution it isurgedthatitdoesinfactsatisfy



s s_ u s e # p d



E R F L O W I N G D O W N A N I N CL I NE D B E D

byarigidco er w hile theopensurface

uiterigorouslyplanebygra ity and

entia lforce . B ut pro idedthebottom

ssofthedimplesand littleroundhollows

ce producedby turbulence( w henthe

seemstopro ethatthemotionmustbe

t wouldbeifthe uppersurfacewere

andfreefromtangentialforce.

bedin~ 3 1ha ingbeendisposedof

ow ta ethe includingcase describedinthe

1 forw hichw eha e assteadyso lution

.. . .. . . ( 57 ,

b ot to m up wa rd s. T hu s ( 7 , ( 8 , ( 9 , ( 1 1 ,

dp

( 3 - y U d .. .. .. ( 5 8 ,

d .

dp

= 2 V d .. .. .. 5 9 ,

-.dy.

d p ( 6

,

d .

V 2 p. .. 6 1 ,

2 -

) = V v . . .. .. .. .. .. ( 6 2 .

ow anysuchsimplepartia lsolutionas

5 3 6 f o r th e s ub - ca s e th e re d e al t w it h a n d we

irtually inclusi e in estigationspecif ied

in ~ 3 8 a ss um e

+ q z ) V . . .. .. .. .. .. .. .. .. .. . ( 6 ) .

tacit lyassumed. Therea lsodiscussionsofproblemsof thistypeby

ethodaregi eninpp. 122-1 8 w ithanaccountofpre ious

a p re i ou s pa rt ( l oc . ci t. N o. 2 ~ ~ 3 A 5 8 P ro f. O r r g i e s

Rayleigh sconclusionthatin theabsenceofv iscositythe

table inthesensethatthisstabilityw oulde istonly

nces cf . supra ~ 27.

Liou il leanaly sis( F ourier Theoriede laC haleur Sturm

i l le s J o u rn a l fo r t he y ea r 1 8 6 a n d Lo r d Ra y le i gh s T h eo r y

o l . I. s h ow s h ow t o e p r es s a n ar b it r ar y f un c ti o n of x , y z b y

lu ti on s of ~ ~ 3 7 3 9 a b o e a nd ~ 4 ( 6 ) , ( 6 7 , ( 7 0 h er e

etherforourpresentcaseor formersub-case thefulfilment

, ( 2 7 , w i t h ou t u si n g th e m et h od o f ~ ~ 3 4 3 5 3 6 .



s s_ u s e # p d



R

tm and V - -m2-q 2...... 64 ;

herefore

d 2~ ? )

+ t [ a + m ( M yC â € ” y2 ] } 2 + tI a( M 2 + q 2 2

Y / ) ~ ~ ~ ~ dy 2

~ y _ C y2 ] ( M + q 2 - to ml 61 0 .. .~ ( 6 5 ,

2 + ( h + I c y+ l ys V = 0 .. . 6 6 .

me

2 + C o + C 4 o4 + & a mp C .. . .. . .. . . 6 7 ;

e r o th e c oe f fi c ie n t of y i i n ( 6 6 w e f in d

( i + 2 ( i + 1 , U C i+ 4 + ( i + 2 ( i + 1 e cif2

l [ ] i ( i - .1 g + f h C i+ k c C i- 1+ l Ci -2 = 0 .. . 6 8 .

e l y i = 0 i = 1 i = 2 . .. a n d re m em b er i ng

s u ff i i s z e r o w e fi n d

e c 2 + h o 0 = 0

2 . ec + 2 . 1 .f c2 ~ h c + k 0 = 0

e o4 ~ 3 . 2 .f c + ( 2 .1 .g + h c + k c + 1 00 = 0

5 . 4 .e c5 + 4 . 3 . f C4 + ( 3 . 2 . g + h C + 1 0 02 ~ l C I = 0

a m p c . & a m p c .. .. .. .. .. 6 9 .

eninorder gi esuccessi ely04 05 06 ... each

c ti o n of c I c 1 0 2 C ; a n d by u s in g i n ( 6 7

ined w ef ind

Ci Ai y + C 2& a mp ( y + c A ( Y .. ... . 7 0 ,

c . a re f ou r a rb i tr a ry c o ns t an t s a n d S , , ~ ,

ho llydeterminate e pressedinaseriesof

hchby ( 68 w eseetobecon ergentfor

spbez ero . Theessentia lcon ergencyof

a s i n ~ 3 9 f o r th e c as e o f no g r a i t y t h at t h e

v = 0 w = 0 i s s ta b le h o we e r s ma l l be E

o.

essthecon ergence. When/ w is

ergenceformany terms butult imate



s s_ u s e # p d



E R F L O W I N G DO W N A N I NC L IN E D B E D 3 3 3

t h e di f fe r en t ia l e u a ti o n ( 6 5 , o r

ducedf romthe4thtothe2ndorder andmaybe

m 7

+ m ( y - c . * . . .. .. .. 7 1 .

o-dimensionalmotion( q = 0 , agreeswith

e pressedinthe laste uationofhis

orInstabilityofcertain F luidMotions

. F eb. 12 1880 . The integra l butnow

nstants( C 0 cl , isst i l lg i eninascending

n d ( 6 8 , w h ic h w i th L = 0 a n d th e t hu s si m pl i fi e d v a l ue s o f e f g p u t in p l ac e o f th e se l e tt e rs b e co m es

1 oc i+ 2 + ( i + 1 i m/ ci + ]

h C i + k i - + c i -2 = 0 .. .. .. 7 2 .

es o f i t hi s g i e s

m c ci = 0 . . . .. . .. . .. . .. . 7 ) ,

ly e ceptinthecaseofoneparticular

. .. . . .. 74 ,

mallerrooto f thee uation

.. . . .. . . .. . . .. . . .. . 75 .

otcon ergenceforva luesofye ceeding

andthustheproofof stability islost.

e uation simplif iedin( 71 forthe

maynodoubtbetreatedmoreappropriately

onofstabil ityorinstabil ity byw ritingit

e no t in g t he t wo r o ot s o f ( 7 5 ] ,

( I - ) } 2 .. .. .. 7 6 ,

- y. _ - y.

alconsiderationoftheinfinities at

. O n e w ay o f do i ng t h is w h ic h I m er e ly

ddonotfo llow outforw antof t ime isto

' -y 2 + o ( ~ - y + & a mp c . ,

- y 2 + C / ( ' - y 3 + & a mp c. ... 77 ,

o ar b it r ar y c on s ta n ts a n d C2 c , . . . c 2 c / . .

edsoas tosatisfythedifferential



s s_ u s e # p d



R

easilydone andw hendoneshow sthat

fora llv a luesofy lessthan4 ande ceeding

f thisindeta ilw ouldbevery interesting

llmathematicaltreatmentofthe

sstream- lines( cur esofsines throughout

w o" cat s-eye borders( correspondingto

w hichIproposedinashortcommunicationto

A ssociationatSw ansea in1880 , " O na

R ay le igh sso lutionforWa esina

. It istoberemar edthatthisdisturbing

mingproofofstabilitycontainedin Lord

( 5 6 , ( 5 7 , ( 5 8 t .

a n d in t er p re t in g t he r e su l t in c o nn e i o n

t h at

hw eha efoundconsistso faw a edisturbancetra e ll inginany ( x , z ) direction o fw hichthe

nthex -directionis-o/m.

4 ) o f ( 7 5 a re v a l ue s of y at p la ce s wh er e

sturbedlaminarf low ise ua ltothexv e locityof thew a e-disturbance.

ounding-planestobeplastic andforce

othofthem soastoproducean

orrugation accordingtotheformula

z ) , thissurface-actionw illcausethroughoutthe

finitesimalwa e-motionifo/misnot

foranyplaneof thefluidbetw eenits

nitycorrespondingtoy= 4 ory= 4 w ill

o /m ise ua ltotheva lueofU forsomeone

oplanesof thef luid andthetrue

he" cat s-eyepattern o f streamlines and

atthisplaneortheseplanes.

ctispublishedinNatureforNo ember11 1880 andin

olumeReportfortheyear.In thisabstractcancelthe

w ithreferencetoacertainsteadymotiondescribedinit

t se e n e t f o o t no t e .

sinreply ( loc. cit. supra thatw henX ciscomple there

sothattheargumentdoesnotfa il regardedasonefor

a luesofw thoughitmaynotcompletelyensurestabil ity .

ectinProc. Lond. Math. Soc. x x v II. 1895 Scientif icPapers

erence inthete t andsupra p. 186 isthattherecannot

ases unlessthisbandofv orticeshasbeenestablished.



s s_ u s e # p d



E R F L O W I N G D O W N A N I N CL I NE D B E D

enmo ingw iththesteady

parallelboundaryplanes e pressed

h wo u ld b e a c on d it i on o f k i n e t ic e u i li b ri u m ( p r o e d

erthe influenceofgra ityandv iscosity and

scositybesuddenlyannulled. Thef luid

brium butisthee uil ibriumstable

letoneorboth bounding-surfacesbe

nyplaceandleft freetobecomeplane

esisofthissurface-operationis

) c os c ot co sm c os q z . .. .. . 7 8 ,

m q ) { c os ( o t- m )

c os q z . .. .. . 7 9 ,

safunctionofmandq w hichimplies

tionstra ellinginoppositex -directions

o ~ o o f ( c o /m t h e wa e - e l oc i ty . He n ce

disturbanceessentia lly in o l ese ll ipticw hirls.

ensteadylaminarmotionisthoroughly

tobrea upintoeddiesine eryplace on

t shoc orbumponeitherplastic plane

egreeofv iscosity asw eha eseen

onstable butthesmallerthev iscosity

fgsinI o rthegreatertheva lueofgsinI

thenarrow erarethe lim itso f this

beenledby purelymathematicalin estigationtoastateof motionagreeingperfectlywiththe

escriptionsofobser edresultsbyOsborne

March15 188 , pp. 955 956 : Thefactthatthesteadymotionbrea sdow nsuddenly

stateof instabilityfordisturbances

ause ittobrea dow n. B utthefact

willbrea downforalargedisturbance

erdisturbance showsthatthereisa

so longasthedisturbancesdonote ceed

fsurprisetometoseethesudden

ssprangintoe istence showinga



s s_ u s e # p d



E R F L O W I N G D O W N A N I N CL I NE D B E D 3 3 7

g ma esitcertainthatifw aterbegi en

eplanesboth atrest andifoneof the

nottoogradually setinmotion andk ept

emotionof thewaterw illbeatf irstturbulent

ofuniformshearingwill beapproached

timateannulmentofthe turbulence.

municationonthissub ecttoSectionA of

nManchester andtoha eitpublished

f thePhilosophicalMaga ine. C orrespondingquestionsmustbee aminedw ithreferencetothe

blem ofaninfinitelylong straight

ginw aterw ithinaninf inite ly longf i edtube.

1888A damsPri ew illbringout

nsonthissub ect.

artero fapoundpers uarefoot( ) istheresistancedueto

aterbetw eentw opara lle lplanes-o facentimetre( 9 ofa foot asunder w henoneof theplanesismo ingre lati e ly totheotherat

tres persecond if thew aterbeatthetemperature0~ C ent.

ca lculatedf romPoiseuil le sobser ationsonthef low of

is1- 4x 10-5ofagrammew eightpers uarecentimetre .



s s_ u s e # p d



T W O - D I M EN S IO N A L W A V E S P RO D U C E D

TINGDISTUR B A NC E .

. E di n . V o l . x x v . r e ad F e b . 1 1 9 04 p p . 18 5 -1 9 6

une 1904 pp. 609-620.

swaterinastraight canal infinitely

w ithvertica lsides. Letitbedisturbed

pressureonthesurface uniformin

atotheplanesides andle ftto itse lf

re . It isre uiredto f indthedisplacementandv elocityofe eryparticleof thewateratany future

onw illbefully specif iedbyagi en

city andagi ennormalcomponentdisplacement ate erypo into f thesurface.

tatadistancehabo etheundisturbed

para lle lto the lengthof thecana l andO Z

etf bethedisplacement-components

componentsofanyparticleof thew ater

onis( x , z ) . Wesupposethedisturbance

wemeanthatthe changeofdistance

ofwateris infinitelysmallincomparison

tance andthe line j o iningthem

directionwhichareinfinitelysmalli n

n.W aterbeingassumedincompressible

ion startedprimarilyfromrestbypressure

isessentiallyirrotational.Hence

d

; = d ( Q , , , 0 t ; = ; = . .. 1 ;

t o r h a s w e ma y wr i te i t f or b r e i t y wh e n co n e n ie n t i s a f un c ti o n of t h e v a r i ab l es w hi c h ma y b ec a ll e d th e

andb( x , z , t isw hatiscommonlyca lled

blemstreatedinthefo llowinggroupofpapersandthe irhistory

ss' O nDeep-WaterWa es byProf. H. Lamb Proc.

. ( 1904 , pp. 3 71-400.



s s_ u s e # p d



R

C o sg 2 + ( P - z ) s i n 4 p2 .. . ( 7

p2

. -- -. 8 ,

e 4p2. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . ( 8 ,

2 a n d 0 = t a n- l / +

changeswhenx passesthroughz ero.

a n d de n ot i ng b y { R D t h e di f fe r en c e

idedby2t w eha eanotherso lutionofour

entf rom( 6 asfo llow s

{ R D 4 + x 4 + ) . .. .. .. .. .. .( 9 ,

+ z ) c si os e -/ - 4t 2( l o

2...........................11 .

m f ig. 1 representsfort= 0theso lutions

, w ithz = 1forcon enience inthe

ichw ef indby ta ingt= 0in( 7 x 2/ 2

t h e mi n us s i gn i n ( 1 0 b e in g o mi t te d f or

x 2 + 2 + ] , [ / x 2 + 2 - _ Z ]

2 4 2 + ) . x ( 1 2

cticalinterpretationof oursolutions

containfullspecif icationsof tw odistinct

neachofw hichq bmaybeta enasa

orasav e locity -potentia l o rasahori ontal

orv elocity orasav erticaldisplacementcomponentorv elocity.Thusweha ereallypreparationforsi

o fw hichw esha llchooseone -= V / 2x ( 7 ,

n.

1 f o r th e wa t er - su r fa c e l e t th e t wo c u r e s o f

splacements ( 12 of thew ater-surface

i enf irst inP roc. R . S. E. J an. 1887 andPhil. M lag.

p. 3 07: ithasnotbeenreprintedhere .

s( loc. cit. thatthe init ia ldisturbance isnotentire ly

sf d doesnotcon erge. Seea lso infra ~ 101.



s s_ u s e # p d



A T E R TW O - D IM E NS I O N A L W A V E S 3 4 1

-̂ ^ - ^ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . - q



s s_ u s e # p d



R

re erywhereatrest.Thedisplacements

t a r e e p r es s ed i n r ea l s ym b ol s b y ( 7 ( 1 0

andby ( 8 , ( 11 w ithafactor/ 2introduced e itherofw hichmaybechosenaccordingtocon enience

asthusbeencalculatedf rom(8 , w ith

f or si v a lu es of t ' 5 1 1 5 2 2 5 an d5 an df or

rofv aluesofx torepresenttheresults

igs. 2and3. E ceptforthetimet= 5

entlyallthemost interestingcharacteristics

atthecorrespondingtime.Thecur e

ptibly lea ethez ero lineatdistances

t if w e c ou l d se e i t i t w ou l d sh o wu s t wo a n d a ha l f

ery interestingcharacterist ics shownin

7be low byw hichw eseethatse era l

lesofordinatesmagnifiedfromone to

nemill ion andtotenthousandmill ion

ibitthemgraphically.

e s f or t = 0 a n d t = ; w e se e t ha t a t

istancesfromthemiddleof the

= 19 andfa llsatlessdistances. A nd

= 0 r e ma i ns a c r es t ( o r p o s it i e m a i m um

foret= 1 w henitbeginstobeaho llow .

istencebeside itandbeginstotra e l

u r e t = 1 w e s ee t h is c r es t t r a e l le d

7 f romthemiddlew here itcameintobeing

s i t h s e e nt h cu r e s ( f i gs . 1 2 w e

9 4 8 6 5 2 2 a t t he t im es 1 X , 2 2 ,

ghtwardsonourdiagramshasits

dua ldowntotheundisturbedle e lat

pe ismuchsteeper andendsatthe

middleofthedisturbance attimes

sometime w hichmustbeverysoon

o llow beginstotra e lrightw ardsfromthe

eshcrest shedofffromthemiddle.At

g ot a s f ar a s x = ' 9 a t t = 2 ~ , a n d 5 r e sp e ct i e l y i t h a s re a ch e d x = 1 7 5 a n d x = 6 7 . Lo o i n g in

nsionofourcur esle f tw ardsfromthe

w ef indane actcounterparto fw hatw e

ontheright. Thusw eseeaninit ia l

calonthetw osidesofacon e crest o f



s s_ u s e # p d



A T E R TW O - D IM E NS I O N A L W A V E S 3 4

eundisturbedle e l sin inginthemiddle

s. Thecrestbecomeslessandless

w n t o he i gh t 1 1 w he n i t be c om e s co n ca e

arw a e-crestsareshedof fonthetwo

y f romitrightwardsandlef twardsw ith

eachremainingfore ercon e . Thusw e

oendlessprocessionsofwa estra e ll ing

ons originatingasinfinitesimalwa eletsshedoffonthe twosidesofthemiddleline. Eachcrestand

increasingve locity . Eachw a e- length

romhollow tohollow becomeslongerand

tw ards a llthisaccordingto law fully

f ~ 3 a bo e .

fnumberspromisedin~ 5abo e

msand magnitudesofthetwoanda

e n x = 0 a n d x = 2 w h ic h t he s p ac e -c u r e f o r

f a il s t o sh o w.

l g= 4 t = 5 - = , s in s( + 0 e P2 .

. 4 C o l. 5 C o l. 6 C o l. 7

o 0

e X e p 2 m '

|

42 1 0000 10-10' 1 57+11-0 196

4 14 0 ' 3 4 4 ' 1 47 8 , , 0 71 7

0

09 - 7541 , 1778-10-10 1891

- ' 8 9 97 , - 0 66 , , , - 8 82

9 - ' 0 0 2 , 3 6 2 , , , 0 016

1 3 7 0 - 8 99 7 , 1 -0 94 + 1 0 -1 0 1- 6 2

0

3 3 8 - ' 5 45 1 , 4 6 6 - 10 -1 0 3 - 24

8 1 26 2 - 2 4 1 , 1 0 - 9 , 3 1 -84

9 ' 7 5 9 1 0 -5 * 0 2 9 6 + 1 0 - 5 02 2 7

1- 09 9 - 89 62 , 2 95 8 , , , 3 1 5 2

00 7 - 68 1 , 5 -7 9 , , , , 4 42 4

92 87 - 49 2 , , 4 5 6 , , 2 6 7

2 0

16 - -68 2 , 212-5 10-144-6



s s_ u s e # p d



R

.

- = 2 si n( + 2 +

4 C o l. C o l. 6 C o l. 7



s s_ u s e # p d






s s_ u s e # p d



E S O N W A T ER [ 3 5



s s_ u s e # p d



A T E R TW O - D IM E NS I O N A L W A V E S

show ninthepre ioustable forthe

tute- ' ; -w eseethat thef irstfactor

w ly f romx= 0tox = oo thesecondfactor

tw een+ 1and-1w ithincreasingdistances

f romz erotozeroasx increases. Thethird

sesgradually f rome- t / hatx =0 to1 at

t h e th i rd f a ct o r is ' 9 9 w hi c h is s o n ea r ly

ofamplitude is fora llgreaterva lues

enby thef irstfactora lone w hichdiminishes

t o 0 at x = o .

gi e n f i gs . 1 2 3 , m a yb e c al l ed

neachof themabscissasrepresentdistancef rom

ance. F ig. 4isatime-cur e( abscissas

= 2h. Itrepresentsav erygradua lrise

f o ll o we d b ya f a ll t o a m in i mu m at t = 2 8 a n d

ns w ithsmallerandsmallerma imum

ions andshorterandshortertimesf rom

t= o . Thesamew ordsw itha lteredf igures

waterle e latany f i edposit ionfarther

ncethan x = 2.Thefollowingtable

100h a llthetimesofz ero lessthan71h

epressionsattheintermediatetimeswhen

5of~ 7 hasitsma imumandminimum

eseele ationsanddepressionsareveryappro imately thegreatestinthe inter a lsbetweenthezeros because

~ 7 v ariesbutslow ly asshow ninthef irst

e.

forthef reesurface in o l esno

aparticularcaseofthe general

ssumptiong= 4 merelymeansthatourunit

acefallenthroughinourunit oftime.

sof~ 3 maybegeometricallye plained

i n g 0 o u r or i gi n o f co - or d in a te s a t a h ei g ht h

anddefiningpasthedistanceofany

t. When asin~ ~ 5-9 w eareonly

the freesurface( thatistosay when

ha t i f x i s a l a rg e m ul t ip l e of z , p x . S e e fo r

f thetableof~ 9. A ndifw eareconcerned

urface w estil lha ep x , if x isa



s s_ u s e # p d



R

p = 1 00 -0 05 .h O = t ar l 1 = 4 51 8 .

TimesofZ ero

andof A ppro imate

i m um A p p ro i m at e Ma i m um

a t io n s an d F P M a i m um E l e a t io n s an d

ons Ele ationand Depressions

Depression

+ ~ 1 4 0 - 7 7 18 5 0 -9 0 + 1 09 1

2 42 0

60 -7478 5 90 - -1058

5- 4 0

~ 1 1 7 * 7 2 4 7 56 7 4 + ~ 1 0 2 5

0 0

1 2 77 6 7 02 5 9 4 5 - ~ 0 9 9

0 75 0

12 7 6 80 6 6 2 0 + 09 6 2

29 0. 8480 40-61 - .1199 * 6595 6451 - -09 3

0. 8219 44- 1 - -1162 * 6 92 6690 + 0904

6807 0

157 -6195 69-21 -0876

3 4 0

sw e

o imation

_gt__

X Z ) CO S 4- + V / x - Z ) S iy : 4 X 2.. 1

resent- f ( insteadof -5 asin% 5-9 ;

1 41 ,

m(1 w ithoutfartherrestrict i esupposit ions. B utifw esupposethatz isnegligibly smallin

andfartherthat

. t 2

. .. ..



s s_ u s e # p d






s s_ u s e # p d



E S O N W A TE R [ 3 5

insteadof+ , isCauchy ssolution ; of

thetimehasad ancedsomuchasto

u i a l en t t o ( 1 5 , " l e m ou e m en t c ha n ge

ppro imation. TheremainderofhisNote

ges ischie f lyde otedtoverye laborateef forts

orthe largerva luesof t. Thisob ect

thee ponentia lfactorin( 8 o f~ 3

ripplingrestrict ionz l . 0w hichv it ia tes( 16

l. I . no te x v i . p. 1 9 .



s s_ u s e # p d



T A N D R EA R O F A F R E E PR O C E SS I O N O F

TER.

. Edin. J une20 1904 Phil. Mag. V o l. v III. O ct. 1904

cationissubstitutedforanother

whichwasreadbeforetheRoyalSociety

y7th 1887 becausetheresulto f that

fectandunsatisfactorybyomissionof

e ferredto in~ 10ofmypaperofF ebruary

nceforthto thelast-mentionedpaperas

e e s u pr a p 3 0 7 .

processionsproducedbysuperpositionofstaticinitiatingdisturbances ofthetypee pressedin

e graphica lly representedby f ig. 1 andleading

n~ 1- , 5-10. Theparticulartypeof

choose isthatchosenattheendof~ 4

butusefulmodif ication maynow w rite

gtX 2

c o / - e . .. .6 ) .

2 2. .. .. .. 1 7 .

z 2 + x 2 , a nd X = ta n- l( x / )

rdv erticalcomponentofthedisplacementofthefluid attimetfrom itsundisturbedpositionatpoint

chmaybeeitherinthefreesurfaceoranyw herebelow

7 , w e h a e f o r th e i ni t ia l h ei g ht o f t he

undisturbedle e l

( ) = .. ... .( 1 8 .

fo rir- tan- / - - sa esconsiderable labourand

ecia llywhen asinourca lculations z =1.



s s_ u s e # p d



R

e asinit ia tingdisturbance arow

ooofsuperposit ionsof ( 18 ; a lternate ly

andplacedate ua lsuccessi edistances

e

o -l i + i , o ...... 19 ,

x + iX , o . ... ... .. 1 9 ) ,

= ( x , 0 - ( x + , 0 . .. .. .. .. .. . 2 0 .

aspace-periodicfunction w ithX foritsperiod.

stitutedfor0 represents-t beingthe

abo eundisturbedle e latt imet in

ncerepresentedby ( 19 .

hate erfunctionberepresentedby

19 impliesthat

P ( x , 0 . .. .. .. .. .. .. .. .. . 2 1 ,

ace-periodicfunctionwithX forperiod.

sthat

- P , 0 . .. .. .. .. .. .. .. ( 2 2 ;

A ndw iththeactualfunction ( 18 , w hich

x , 0 , t he fa ct th at & l t b ( x , 0 = q ( - x , 0

- 0 . .. .. .. .. .. .. .. .. .. . ( 2 ) .

of thecharacterfig.5 symmetricaloneach

andminimumordinate . TheF ourier

, 0 , w he n su b e ct t o( 2 2 a nd ( 2 ) , g i e s

7r

c os -- + A c os 3 - + A c os 5 + .. .( 2 4 .



s s_ u s e # p d



N T A N D RE A R O F A F R EE P R O C E SS I O N 3 5

unctionsgeneratedbyadditionof

ore uidifferentarguments. Letf x ) be

periodicornon-periodic andlet

i X ) . .. .. .. .. .. .. .. .. . 2 5 ;

x ) = P ( x + X ) . . .. . .. . .. . .. .. . .. . .. 2 6 .

c e p a ns i on o f P ( x b e e p r es s ed a s

c os a + A c os 2 a + A c o s 3 a + w h er e ac = 2

s i n 2c a+ B 3 s in x + . .. X . .. .. . 2 7 .

er w eha ebyF ourier sana ly sis

.2 .r . . 2

c l O . 2 7w - C 00 . 27 r X

+ iX ) c os d f x ) cos

- 0 .2 7 rr

x + iX ) sin d f x ) sin2

- o X . . .. .. 2 9 .

, a sb y( 1 9 ) , ( 2 0 ,

0 -P X + X O ) . ......... 3 0 .

stoz ero reducestheA stoz erofore en

or o dd v a l ue s of j g i e s i n v i r tu e of ( 2 2 ,

, 0 cos l . ( 3 1 .

4 ( 6 , ( 1 2 , a bo e a nd a cc or di ng to th e

e

SI 2 ( P / x ) . ... p. . . ( 3 2 .

c e p an si on ( 2 4 o f P ( x , 0 , w e ha e

. 2w r 4 ~ S RS + C X V 2 co .2 wr

cos - X R d x - cs

Z + t X ) . . .. .. 3 3 ) .

2



s s_ u s e # p d




ast memberofthise uationfacilitates

negral. Insteadofcos inthe last

. 2 w ~ ~ ~ ~ ~ ~ 2 7 r . 27 -r : 2 77 - V 2 n

( 3 4 .

snodifferenceinthe summationf__ d ,

ppearsforthe samereasonthatthe

sa p pe a r be c au s e of ( 3 0 . T h us ( 3 3 ) b e co me s

n L

3 5 ;

+ t )

+ t X ) = t o

- an dt = â € ” z . ( 3 6 .

w emayomitthe instruction{R S because

sintheformula:thuswefind

2i r z 2 r 8V 2 _ 2

fd ae 2 2 C A

ismadeinv irtueofLaplace scelebrated

W

allowsusreadily toseehow neartoa

a ph o f P - x 0 f o r an y p ar t ic u la r v a l u e of

4 r

1 A 5/A = V 3 .e.

= 4 ; w e ha e

14 A /A = 0 24 95 A / A = - 0 3 4 7. .. 3 9 .

ut_ ofA ; andA about-LofA .

nusoidality butnotq uitenearenough

Tr y n e t X = 2 ; w e ha e

" = 0 0 18 67 A / A = - 00 10 78 .. . 4 0 . \ /



s s_ u s e # p d



N T A ND R E AR O F A F R EE P R O C E SS I O N 3 5 5

ndthofA1 andA about1~ x 10-6of

enoughappro imationforourpresent

bleinany ofourcalculations:A is

eptibleifincludedin ourcalculations

ofoursignif icantf igures : butitw ould

ourdiagrams.Henceforthweshall

yw iththef reesurface andta ez = h the

fcoordinatesabo etheundisturbedle e l

ceatany timetafterbeingleft

cedaccordingtoanyperiodicfunctionP( x )

s e as i n ( 2 7 ; t a e f i rs t f o r th e i ni t ia l

cement a simplesinusoidalform

. . .. . .. . .. . .. . .. . . 4 .

( 3 ) , a n d ( 4 a b o e l e t w ( z , x , t b e t he d o wn w ar d s v e r t ic a l co m po n en t o f di s pl a ce m en t . W e t hu s h a e a s t he

themotion

.. 42 ,

4 ) .

- c c o s t /g n . .. . .. . .. . .. 4 4 ,

l- nownlawoftwo-dimensionalperiodic

w ater. A ndformula( 44 w ithC e-m= A

h( 41 . Hencetheadditionofso lutions(44 ,

h A s uc c es s i e l y pu t e u a l to A , A 2 . . .

i t hc = 0 fo r th e A s a nd = - r f or t he B ' s g i e s

ertica lcomponent-displacementatdepthz -h

timet= 0thew aterw asatrestw ithits

g t o ( 2 7 . T h us w i th ( 3 8 , a n d ( 4 4 , w e

a n d ( 2 7 , a n d p u tt i ng m = 2 7 r/ X , w e se e

nduetoanyoneof theA sorB ' sinthe

ndlessinfinite rowofstandingwa es

e u a l to X / a n d ti m e- p er i od s e p r es s ed b y

4 5 .

..........



s s_ u s e # p d



R

riodic becausetheperiodsofthe

e ingin erse lyas/ , a renotcommensurable .

2 h a s pr o po s ed i n ~ 1 7 w h ic h a c co r di n g to

s f o r th e f re e s ur f ac e o n ly a l i tt l e mo r e th a n 1/ 1 00 0

aranapproachtosinuso ida lity thatinour

dthemotionas beingperiodic with

1 . Th is m a e s r = V / 7 wh en a s in ~ 5 w e

~ 10 simplif yournumerica lstatements

d h = 1 w hi c h ma e s t he w a e - le n gt h = 2 .

o f " f rontandrear " remar now

parallel straightstandingsinusoidal

startede erywhereo eraninf initeplaneof

er maybeidea lly reso l edintotwo

w a esofha lf the irhe ighttra e ll ingin

ua lv e locit ies2/ /7r.

gthew holesurfacewithstandingw a es

ati esideof the line( notshownin

, thatisthe le f tsideof0theoriginof

ea ethew aterplaneandmotionlessontheright

distancesonthe le f tsideof0 there

standingw a ese ui a lenttotw otra ins

o f w a e - l e ng t h 2 t r a e l li n g ri g ht w ar d s an d

//7r.Thesmoothwaterontheright

dedby therightwardprocession.

pro esthatthee tremeperceptible

ession(mar edR inf ig. 10below does

R onthe le ftsideof0 broadeningwith

therighto f0 perceptiblydisturbthe

lsopro esthatthesurfaceat0has

roughalltime.It farthershowsthat

sveryappro imately sinuso ida l w ith

throughaspaceOF ( f ig. 9 to theright

me andthat atanyparticulardistance

appro imationbecomesmoreandmore

ances. WhatIca llthef ronto f the

isthew a edisturbancebeyondthepo intF ,

stancerightwardsfrom0 wherethe

idalityo f shape andsimpleharmonic



s s_ u s e # p d




nly j ustperceptiblyatfault. Wesha llf ind

esare asshowninf ig. 9 lessandlesshigh

atgreaterandgreaterdistancesfrom 0

butthatthew a e-he ightdoesnotat

bruptlytonothing.Thepropagational

ofthe disturbanceisinrealityinfinite

terasinfinitely incompressible.

efrontoftherightwardprocession

o llow ingitf rom0 issimplygi enby the

ev a luesofx , o f themotionduetoaninit ia l

ofsinusoidalfurrowsandridgesonthe

esentsastatic initialconfiguration

x , 0 , appro imately rea lisingthecondit ion

epresentsonthe samescaleofordinates

atthetime25r inthesubse uent

conf iguration w hich forany timet w e

d ef in ed as fo ll ow s: Q ( x , t = ( ( x , t - ( x + 1 t + ( x + 2 t - .. .a di nf .. .. 4 6 ,

d ef i ne d b y ( 1 7 , w i th z = 1 a n d g = 4 .

ata lldistancessofarle f tw ardf rom0

earoftheleftwardprocessionhasnot

particulart ime t a f terthebeginning

t o f ~ 1 c a lc u la t ed a c co r di n g to ~ 1 8 1 7

il lmere ly standingwa es idea lly

dandleftwardprocessions. LetI

offig.10 bethepointof theideally

otprecise lydef ined w herethe le f tward

rtime t becomessensiblyinfluenced

nIandRthe wholemotionistransitional

egularsinusoida lmotionP( x , t o f the

toregularsinuso idalmotionofw a eheight2P( x , t , f romR to0 andontoF o f f ig. 9 thebeginning

anceintherightwardprocession.Hence

wardprocessionfromthewholedisturbancedue totheinitialconfiguration weha eonlytosubtract

m Q ( x , t c al c ul at ed f or n eg at i e v a l ue s of x . T h us

holeofthe leftwardprocessionis

, t f or ne ga ti e v a lu es of x . .. .. . 4 7 .



s s_ u s e # p d



R

esurface thusfoundfortheleftward

25r.

t , w h ic h a pp e ar s i n ~ 1 a s a n it e m

m mi n g sh o wn f o r P ( x , 0 i n ( 1 9 ) , a n d

t a t t he e n d of ~ 1 , a n d wh i ch h a s be e n us e d

nsforQ ( x , t ; isrepresentedinf igs. 6

n d t = 2 5 r re s pe c ti e l y.

hepo intsof f ig. 6 representing

ca lculationhasbeenperformedso lely forintegra l

atf irstscarce ly tobee pectedthatafa ir

uldbedrawnfromsofew calculatedpoints

ctua llybeendraw nbyMrWitheringtonw ith

anthesepoints e ceptinformationastoa ll

ngthe lineofabscissas , throughthew holerange

ulatedpo intsaremar edoneachcur e:

w iththek now ledgeof thezeros the

ryclose ineachcasetothatdraw nby

x , t , f o r po s it i e i n te g ra l v a l u es

sedby thefo llow ingarrangementsfora o iding

mmationofa sluggishlycon ergent

4 6 , b y u se o f o ur k n o wl e dg e o f P ( x , t .

a nd ( 1 9 ,

0 t - ( 1 t + ( ( 2 t -... ad. inf.... 48 ,

( - ) i ( i t . .. .. .. .. .. .. .. ( 4 9 .

- i t = b ( i t ,

0 t . .. .. .. .. .. .. .. .. .. .. 5 0 .

4 6 , w e ha e

x , t - ( + l t + ( + 2 t - ( c + 3 , t + ...

( + 1 t - ( + 2 t + ( + 3 , t -...

Q ( x , t = g[ ( , t - ( x + 1 t ] = = D( x , t ... 5 1 .

t ionsof thise uation w ef ind

( - i2Q ( x , t - -1 iD , t + ... + D x + i - t ...... ( 52 .



s s_ u s e # p d






s s_ u s e # p d



N W A TE R [ 3 6



s s_ u s e # p d



0 , andaportionof thecur eofsineswhichveryappro imatelyagreesw ithitatgreatle f twarddistances.



s s_ u s e # p d



o f r / G o f 1 f

nto f rightwardprocession. Graphof ( 46 fort =25T r.



s s_ u s e # p d



N T A ND R E AR O F A F R EE P R O C E SS I O N 3 6

_ _

-4 -I I

~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

i



s s_ u s e # p d



R

0 a n d us i ng ( 5 0 , w e fi n d fi n al l y

) i p( 0 t - ( -y1 D ( 0 t + ... + D( i -1 t ( 5 ) .

nienttocalculateQ ( 1 t , Q ( 2 t ...

tt ingthepointsshow ninf ig. 9.

dofassumingasin( 47 theca lculation

ne g at i e v a l ue s o f x , a v e r y tr o ub l es o me a f fa i r w e

us . W e h a e b y ( 4 6

t- 4( x + 1 t + ( - + 2 t -...

t - ( - + l t + ( -+ 2 t -....

( - t = ( x , t - ( x + 1 t + ( x + 2 t -...

- + 2 t -. ... ... .. ( 5 4 .

usedinthef irsttermof ( 54 , thatits

s it i e a n d ne g at i e v a l ue s o f x , w e h a e

( ( x - i t . He nc e( 5 4 m ay be wr it te n

( - x , t = 2 ( -1 i & gt ( x + i t = P ( x , t ( 55 .

t = P ( , t - Q ( x , t . .. .. .. .. .. .. .. 5 6 .

w e fi n d

t . .. .. .. .. .. .. .. .. .. .. 5 7 ,

aterdueto theleftwardprocession

ancex f rom0onthe le f tside x be ing

integra lorf ractional. Ha ingpre iously

t f o r po s it i e i n te g ra l v a l u e s of x , w e h a e f o u n d

atedpo intsof f ig. 10forthe le f twardprocession.

o ingplansdescribedin~ 11-28

ymeansforunderstandingandw or ingoutin

mt= 0tot= co o fagi enf initeprocession

hsuchdisplacementof thesurface andsuch

w thesurface astoproduce att= 0 a

rmorew a esad ancingintostillw ater

sti l lwaterintherear. Toshow thedesired

tendf ig. 10le f tw ardstoasmanyw a e- lengths

epoint I describedin~ 24. In ertthe

e ly torightandle f t andf it itontothe



s s_ u s e # p d




tendedrightwardssofarastoshow noperceptible

00 or3 00 o foursca le . Thediagramthus

hewatersurfaceattime25rafter acommencementcorrespondinglycompoundedfromfig.8 andanother

esentther earofthefinite( two-ended

owconsidering.

heproblemthusindirectly so l ed

f1000w a e-crestsinthebeginning the

on

- i t . .. .. .. .. .. .. .. 5 8 ,

undaccordingtotheprinciplesindicated

pressthesamesurface-displacementasour

andtheproperve locit iesbe low thesurface

arightw ardprocessionofwa es. Ourpresent

ytheinitialsinusoidalityof thehead

nfiniteprocession tra ellingrightwards

thehydro ineticcircumstancesofaprocessionin adingstil lw ater. Ourso lution andthe itemtow ards

nd7 andinf ig. 2of~ 6abo e show how

med.Thewholein estigationshows

nganydef inite" group- e locity w eare

oupof tw o three four oranynumber

a es. Ihopeinsomefuturecommunication

inburghtoreturntothissub ectin

yprinciplesetforth byO sborne

heinterferentia ltheoryofSto estandR ay le igh+

f initegroup- e locity inthe ircaseofan

ysupportinggroups.B utmyfirst

eperformanceofw hichIhopemaynotbe

f imypromisesregardingship-wa es and

ingina lldirectionsf romaplaceofdisturbance inw ater.

showsomeofthe mostimportant

enca lculated andw hichmaybeuseful

esub ectofthepresentpaper.

i . 18 77 p p. 3 4 - 4.

r C a rm b .U n i . C al e nd a r 1 8 76 .

o l . i. 1 8 77 p p . 24 6 -7 .



s s_ u s e # p d



A V E S O N W A TE R 3

= , , / p+ Z D x , 0 = ob x , ) - o X + 1 0 .

D( x , 0 [ x ' o X , O 0 I D( x , O )



s s_ u s e # p d




~ ~ o E~ 2 2 25 D1 2T

' - 00 02 * 0 00 0 + 00 01

4 1 * 0 00 5 - 0 0 01 - ' 0 00 2

1 - 0 01 1 + 0 00 1 - - 0 0 0 2

2 0 02 4 ~ 0 0 0 + 00 06

' 0 04 4 - - 0 00 + 00 20

0 0 75 - - 0 02 - . 00 1 8

' - 0 11 8 - - 0 0 05 - - 0 05 5

' 0 1 74 ~ 0 0 5 0 + 0 1 17

0 0 24 6 - - 0 06 7 - ' 0 1 6

- 0 3 3 + - 0 06 9 + ' 0 14 6

- 04 4 - ' 0 07 7 - ' 0 18 8

- 0 55 0 + - 0 11 1 + 0 28 1

- 06 79 - ' 0 1 70 - ' 0 8 6

- 0 82 0 + 0 2 16 + 0 7 7

-0917 - -0161 - . 0101

7 - 11 1 - - 0 06 0 - ' 0 7 2

P 1 2 99 + 0 1 2 + 0 55 8

6 - 1 47 2 - - 0 24 6 - - 0 0 2

5 0 - 16 51 - - 02 14 - ' 0 62 6

4 9 - 1 8 2 + 0 4 12 + 02 6 7

9 - 20 16 ~ 0 14 5 ~ 06 7

' - 22 01 - - 04 92 - ' 0 26 6

- 2 8 5 - - 0 22 6 - ' 0 7 1

4 - 2 56 9 ~ 0 4 8 7 + 0 0 21

7 * 2 75 2 + 0 46 6 ~ 0 7 28

* 2 9 4 - 0 2 62 + , 0 4 25

8 - 1 12 - - 06 87 - ' 0 41 0

* 3 2 8 7 - - 0 27 7 - - 0 7 51

- 4 59 + 0 47 4 - ' 0 29 0

* 3 6 29 + ' 0 76 4 + 0 4 4

- 7 94 + -0 3 0 + ' 0 7 41

* 3 9 5 6 - 0 41 1 + 0 42 9

- 4 11 2 - 0 8 4 0 - 1 0 1 9 0

* 4 2 6 7 - - 06 50 - ' 0 64 2

- 4 41 6 - - 0 0 08 - - 0 6 5 7

- 4 56 0 + 06 4 9 - ' 0 2 82

- 47 02 - + 0 9 1 + 0 22 4

4 8 40 + 0 70 7 + 0 5 82

- 49 7 + 01 25 + 06 4

- 5 10 1 - - 0 5 18 ~ , 0 4 17

5 22 6 - - 09 5 + ' 0 0 5

8 * 5 4 8 - ' 0 97 0 - -0 3 2

- 54 64 - - 06 8 - - 05 56

5 5 80 - 0 0 8 2 - ' 0 5 78

- 5 69 0 + 0 4 96 - ' 0 4 21

- 5 79 7 + 0 9 17 - ' 0 1 52

- 5 90 0 + 1 0 6 9 + 0 1 41

- 60 01 + 0 92 8 ~ ' 0 7

4 ' - 60 98 + -0 55 5 + ' 0 50 1

- 61 9 ~ 0 05 4 + 0 5 06

* 6 28 4 - ' 0 45 2 + ' 0 40

5 * 6 7 2 - ' 0 8 5 5 + 0 22 6

- 6 45 9 - - 1 08 1 + 0 0 2 2

6 54 0 - ' 1 10



s s_ u s e # p d



-WA V ES.

gsof theR oyalSocietyofEdinburgh J une20 1904

u n e 1 90 5 p p . 7 3 - 7 57 .

p- a es.

ew hatcumbroustitle" Tw o-dimensiona l "

" C a n al W a e s t o d en o te w a e s i n

abottom andv erticalsides w hich if

irsource becomemoreandmore

ensionalatgreaterandgreaterdistances

presentcommunicationthesourceis

ontwo-dimensionalthroughout the

pecti elyperpendiculartothebottom

ofthecanal:thecanal beingstraight.

p inthepresentcommunicationand

1- 1 isusedforbre ity tomean

epthatthe motiondoesnotdiffer

dbe ifthewater beingincompressible

onditionispracticallyfulfilled in

edistancebetw eene erycrest( po int

n , andneighbouringcrestone itherside is

hirdof itsdistancefromthebottom.

e s I m e an a n y wa e s p ro d uc e d in o p en

inggenerator andforsimplicity I

generatorto berectilinealanduniform.

p floatingonthewater orasubmarine

uniformspeedbelow thesurface or

notincludean interestingclassofcanalwa esofwhich

firstgi enbyK ellandintheTrans.Roy.Soc.Edin. for

chthewa elengthisvery longincomparisonw iththedepth

andthetrans ersesectioniso fanyshapeotherthan

nta lbottomandv erticalsides.



s s_ u s e # p d



WA V ES

anelectrif iedbodymo ingabo ethe

wa es if themotionof thew aterclose

dimensional theshiporsubmarine

ngitssides( orasubmergedbarha ing

ingtothesidesof thecanal w ithf reedom

Thesubmergedsurfacemustbecy lindric

endiculartothesides.

rcylindricbarof diametersmall

below thesurface mo inghori ontally

amathematicalproblemwhichpresents

worthyofseriouswor foranyonewho

t. Thecaseofaf loatingpontoonismuch

eofthediscontinuitybetweenfreesurfaceof

ressedbyarigidbody ofgi enshape

sierproblemthaneitherof those I

aoraforci econsistingofagi encontinuous

thesurface tra e ll ingo erthesurface

understandthere lationof thistothe

inetherigidsurfaceofthe pontoonto

imagineappliedto it agi endistributionII

ereperpendicularto it. Ta e0 anypo intat

undisturbedw ater- le e l draw OX para lle l

andOZ v erticallydownw ards. Let

nt-componentsofanyparticle ofthewater



ofwateris infinitelysmallincomparison

tance andthatthe line j o iningthem

directionwhichareinfinitelysmalli n

n. F orliberalinterpretationofthis

w . Waterbe ingassumedfrict ionless its

yfromrestbypressureapplied tothe

llyirrotational.B utweneednotassume

mediate ly thatit ispro edbyour

w heninthemw esupposethemotiontobe

av eryusefulw ordintroduced a f tercarefulconsultationw ith

mybrotherthe lateProf. J amesThomson todenoteany



s s_ u s e # p d



R

ionsofmotion w henthedensityof the

ty a r e

d . ( . .. .. .. .. .. .. .. ( 5 9 ,

e o f gr a i t y an d p t he p r es s ur e a t ( x z t .

dtobe incompressible w eha e

.. . .. . . .. . . .. 60 .

sumedtobe infinitesimal thesecond

tmembersof ( 59 arenegligible and

become

ferenceoftwodifferentiations gi es

timethemotionis z eroorirrotational

e er.

srotationalmotionin anypartof

stingtok now w hatbecomesof it . Lea ing

trestrictiontocanalw a es imagine

moothsea inaship k eptmo inguniformly

-ropeabo ethew ater. Loo ingo erthe

erofdisturbedmotion showingbydimples

littlewhirlpools.Thethic nessof

othingperceptiblenearthe bowto

arthestern moreorlessaccordingto

moothnessof theship.Ifnowthe

scosityandbecomesaperfectfluid the

otionte llsusthattherotationallymo ing

e ship andspreadsoutinthe more

andbecomeslost ; w ithout how e er

mecerta inthatifanymotionbegi enw ithinaf initeportion

ble li uidorigina llyatrest itsfate isnecessarilydissi



s s_ u s e # p d



WA V ES

whichbecomesreducedtoinfinitely

f inite ly largeportionof li u id. Theship

lm seawithoutproducinganymore

dstern butlea ingw ithinanacuteangle

smoothship-w a esw ithnoeddiesor

The idea lannulmentof thew ater s

siderablythetensionofthetow-rope but

thasstillw or todoonane erincreasing

ese tendingfartherandfartherright

r ea o f 1 9~ 2 8 ( t a n- / - o n e ac h s id e o f

alsee inabout~ 80below . R eturningnow

oandcanalw a es: w e inv irtueof

. .. .. . .. . .. . 6 )

t iscommonlyca lledthe" v e locity -potentia l ;

ie n t w e s h al l w ri t e in f u ll ( ( x , z , t . W i t h

esby integrationw ithrespecttox andz,

.. . .. . .. . .. . .. ( 6 4 .

d + = 0. . .. . . .. . . .. . . .. . . .. . . .. . . 65 .

m e th o d t a e n o w

= - k e -n s in m ( x - v t . .. .. .. .. .. . 6 6 ,

nde pressesasinusoida lw a e-disturbance

tra e ll ingx -w ardsw ithv e locityv .

y -pressure I w hichmustactonthe

motionrepresentedby ( 66 , w henmn v , k

ap p ly ( 6 4 t o t he b o un d ar y . Le t z = 0 b e t he

n d le t d d en o te i t s de p re s si o n a t ( x , o t ,

thatistosay

= d f( x , z , t z = = m sin m( x - v t ... 6 7 ,

sw ithinf inite ly smallv e locit iese erywhere w hilethe

ainsconstant.Aftermanyyearsoffailure topro ethat

Helmholt circularringis stable Icametothe conclusionthatitis essentiallyunstable andthatitsfatemust betobecomedissipated

thisconclusionbye tensionsnothithertopublished

ribedina shortpaperentitled:" O nthestabilityof

dmotion inthePhil. Mag. forMay1887. [ R eprinted



s s_ u s e # p d



R

hrespecttot

. .. . .. . .. . .. . .. . .. . . ( 6 8 .

urface wemust ing , putz = d andin

becaused k , are infinitely smallquantit ies

eirproductisneglectedin ourproblemof

ts. Hencew ith( 66 and( 68 , andw ith

ace-pressure ( 64 becomes

t = g k c os m( x - v t - I + gC... 69 ;

raryconstantC ta en= 0

m x - t . .. .. .. .. .. .( 7 0 ;

( 6 8 , w e h a e f i n a ll y

.. . .. . .. . .. . .. . .. . 7 1 .

/gm weha eI= 0 andthereforewe

usoida lw a esha ingw a e- lengthe ua lto

nownlawofrelationbetweenv elocity

eaw a es. B utif v isnote ua lto / g/m

sw ithasurface-pressure( g-nm 2 dw hich

thedisplacementaccordingas

g /m .

be:-gi enI asumofsinuso idal

s in g le o n e a s i n ( 7 0 ; - r e u i re d d t he

f thew ater-surface. Weha eby ( 71

operlyalterednotation

- v t+ ) . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . ( 72 ,

v t + ) + A co sf ( - t + y ( 7 ) ,

2

r e gi e n c on s ta n ts h a i n g di f fe r en t v a l u es i n t he

s andv isagi enconstantve locity .

e presses withtwoarbitraryconstants

of f reew a esw hichwemaysuperimposeonany

andinstructi e inrespecttothe

s toapply ( 72 toaparticularcaseof



s s_ u s e # p d



WA V ES. 7

nofperiodicarbitrary functionssuchasa

onstantpressures andz eros one ua l

a e ll ingwithve locityv . B utthismustbe

t to letusgetonwithship-w a es and

a e a s a c as e o f ( 7 2 , ( 7 ) ,

s 0+ e 2 c os 2 0 + e tc . = g c 2 2 c + e

e co s 0 + e 2 . .. . .. . .. 7 4 ,

o - c os + - 2 c o s0 + e t c. .. . .. . .. . 7 5 ;

- v t + / ) . .. . .. . .. . .. . .. . .. . .. 7 6 ;

g a . . _ a. ( 7 7 ;

.. . .. . . .. . 7

c & l t 1 . R em a r t h at w h en v = 0 J = o o

a n d ( 7 4 , d = l / g w h i ch e p l ai n s ou r u ni t o f

amicalconditionsthusprescribed

remar f irstthat( 74 , w ith( 76 ,

icdistributionofpressureonthe surface

yv ; and( 75 representsthedisplacement

resultingmotion whenspace-periodic

sthe surface-pressure. Anymotion

uentonany init ialdisturbanceandnosubse uentapplicationofsurface-pressure maybesuperimposedonthe

75 toconstitutethecompleteso lution

e motioninwhichthesurface-pressure

.

roughlytheconstitutionofthe

f o r 1 i t i s he l pf u l to k n o wt h at n d e no t in g

e i n te g er w e h a e

e 2 co s 20 + e tc . = S b7 + ( x n a ( 7 8 ,

/ e .

9 .

oo



s s_ u s e # p d



R

15abo etotheperiodic function

dmemberof ( 78 .

membersof( 78 isillustratedbyfig.11

9 ( C

e = 5 an d c on s e u e n t ly b y ( 7 9 , b / a = 1 10 ;

esentsthefirstmember andthetwolight

rmsof thesecondmember w hichareas

agramallowstobeseen onit.There

mentbetweeneachofthe lightcur es

ycur ebetw eenama imumandthe

t. Thusw eseethate enw itheso

e a n o t v e r y ro u gh a p pr o i m at i on t o e u a li t y

fperiodsof thefirstmemberof ( 78 anda

member. If e is& lt 1byaninf inite ly

imationisinfinitelynearlyperfect.

9thatfig.12 cannotshowany

asca leofordinatesone-tenthof thato f f ig. 11.

ntbetweenthefirstmemberof( 78 and

memberwithv aluesofeapproaching



s s_ u s e # p d



WA V ES

e followingmodificationofthelast

e 2 y - ( 1 - e2

- C 1 - e 2 + 4 e s in 2 0"

* 8 9 O 9

~ ~ F

IIisv erygreatwhen0is v erysmall

ess0isv erysmall( orv erynearly= 2i7r .

e

. .. . .. . .. . .. . . * ( 8 1

gIIappro imate lybyasingletermof the

.

a lso lution( 75 ; andremar that

e te r m of ( 7 5 i s i nf i ni t e o f wh i ch t h e

arin( 70 . Hencetoha ee ery term

st h a e J = j + 8 w h er e j i s a n i n t eg e r an d

m a y co n e n ie n tl y w ri t e ( 7 5 a s f ol l ow s :

e c o s j 0

1 + j l + +-2 +

0 e + 2 cos ( j + 2 0 _ a di nf. . . 8 2 ;

. . . . . .. . . .. . . .. . . .. . . .. 8 ) ,

initeandinf initeseriesshow nin( 82 .



s s_ u s e # p d



R

e 8 = - a n d in t h is c a se J c a n be

s asfo llow s. F irstmultiplyeachtermby

nd we fi nd

e + [ cos ( j + 1 0 + 2 cos ( j + 2 0 + etc.

d e e- 8c os ( j + 1 + e l- co s( j + 2 0 + e tc .

+ ' de e- { RS q j + l ( 1 + e + e 2 2 + etc. ;

a n d a s i n ~ 3 a b o e { R S d e no t es

lf sumfor+ t. Summingthe infinite

fde forthecase8= a w ef ind

+ ( { R S q + 2 l og .. .. .. .. .. .. .. . 8 4 ,

+ l o g1 + V e c o s 0 + e s in 1 0

2 & g t 2 = 2 t a n2 1 mu -2 10 . 8 5 ,

In - t/ esin0

o g

+ l g 1 - 2 e c os 1 0 e

+ t a n 2 - e8 i n ( 8 6

o f t- ( 8 2 g i e s

1 - e 0 e c o s.

8 ) g stanhesoinofo

c os ( j 0 lo g - 2 ec os e

0 ta n- 2 es in .( 8 6 .

o f 8= ~ , ( 8 2 g i e s

C os

+ . .. +

p r es s ed ( 8 ) g i e s t he s o lu t io n o f ou r

o f ~ ~ 4 6 -6 1 I h a e t a e n e = ' 9

ineticil lustrationsinLectureX . o fmy

p. 11 , 114 f romw hichf ig. 12 andpartof



s s_ u s e # p d



WA V ES

e n . Re su lt s ca lc ul at ed f ro m ( 8 ) , ( 8 6 , ( 8 7 ,

- 1 6 a l l fo r t he s a me f o rc i e ( 7 4 w i th

rdifferentv e locit iesof itstra e l w hich

s 2 0 9 4 0 o f j . T h e wa e - le n gt h s

th e se v e l oc i ti e s ar e [ ( 7 7 a b o e 2 a /4 1

. Theve locit iesare in erse lyproportiona l

Eachdiagramshow stheforci ebyone

f fig. 12 andshow sbyanothercur ethe

ew ater-surfaceproducedby it w hentra e ll ing

speeds.

be ingthehighest o f thosespeeds

forci etra e ll ingatthatspeedproduces

tupwardswherethedownwardpressureis

umdownwarddisplacementwherethepressure

ard isleast. J udgingdynamica lly it iseasy

eaterspeedsoftheforci ewouldstill

e themeanle e lw herethedownw ard

sgreatest andbelow themeanle e lw here

inishingmagnitudesdow ntoz erofor

e f or a ll p os it i e v a l ue s of J & l t 1 a s er ie s

houghsluggishlyw hene 1 , byw hichthe

actly calculatedfore eryva lueof0.

f o r wh i ch J = 4 4 a n d th e re f or e b y

a /9 7r a nd X = a /4 5 . Re ma r t ha t th e sc al e of

only1/2 5of thesca le inf ig. 16 andseehow

ter-disturbancenowincomparisonwith

meforci e butthreetimesgreaterspeed

a e - le n gt h ( v = \ / g a/ 7 r X = 2 a . W i t h in

w eseefourcompletew a es v ery

al betweenM M tw oma imumsof

oste actly (butv eryslightly lessthan

sbetw eenC andC . Imaginethecur etobe

ghout andcontinuedsinusoidallytocut

nC C atra inof4-sinusoida lw a es

edthroughouttheinfiniteprocession...CC...weha eadiscontinuousperiodiccur emadeupof

44periodsofsinusoidalc ur ebeginning



s s_ u s e # p d



S O N W A T ER



s s_ u s e # p d



WA V ES

~ ~ b

~ ~ ~ ~ ~



s s_ u s e # p d



P W A T ER S HI P -W A V E S 3 8 1

D



s s_ u s e # p d



R

hechangeateachpoint ofdiscontinuity

hangeofphase.Aslightalteration

ew ithin60~ oneachsideofeachC

tinuouswa ycur eof f ig. 15 w hich

aceduetomotionofspeedV ga/9wrofthe

entedby theothercontinuouscur eof

8isapplicableto f igs. 14and1 e cept

forci e w hichisV / a / 197rforf ig. 14and

andotherstatementsre uiringmodification

" w a e s " i n r es p ec t t o fi g . 15 s u bs t it u te

nd20~ inrespectto f ig. 1 .

def iningMMinrespectto f igs. 15 14

thecaseof f ig. 1 .

at assa idin~ 48 theformula

, ( 8 7 } g i e s fo r a wi de r an ge o f ab ou t 12 0~ o n e ac h

1 80 . si n( j + ) 0 .. .. .. .. . 8 8 ,

8 49insymbols itbe ingunderstoodthatj

4 andthate is99 oranynunericbetw een

uldgi eashortansw ertothisq uestion

neticideas Here istheonlyanswerI

andseehow intheforci edef ined

re isa lmostw holly confinedtothespaces

sideofeachof itsma imums andisv erynearly

3 00~ . It isob iousthatif thepressure

theselast-mentionedspaces whilein

neachsideofeachma imumthepressure

74 , theresult ingmotionw ouldbesensibly

ewerethroughoutthewholespace

6 0 ~ ) , e a ct ly t ha t gi e n by ( 7 4 . H en ce w e

oughnearly thew holespaceof240 f rom

lmoste actly sinusoida ldisplacementofwatersurface ha ingthew a e- length3 60~ / j + 2 duetothetranslationa l



s s_ u s e # p d



WA V ES

te pectsosmalladif ferencefrom

wh o le 2 4 0~ , a s c al c ul a ti o n by { ( 8 ) ( 8 6 ,

d a n d a s i s sh o wn i n f ig s . 18 1 9 2 0 b y t he

ndsideofC w hichrepresentsineach

- - d ( 180 .sin j + ) . ..... 89 ,

0 f romonecontinuoussinuso ida lcur e .

ssofthisdifferencefordistancesfrom

0~ , andthereforethrougharangebetw een

0 ~ , i s v e r y r e ma r a b le i n e ac h c as e .

p re t at i on o f ( 8 8 a n d fi g s. 1 8 1 9 2 0

e so l ut i on t 8 ) , ( 8 6 , ( 8 7 } a " f r ee

d in g to ( 7 ) , t a e n as

. s in ( j + ) . .. .. .. .. .. .. ( 9 0 .

ulstheappro imatelysinusoidalportion

nf igs. 1 , 14 15 andappro imately

elysinusoidaldisplacementinthecorrespondingportionsofthespaces CC andCConthetwo sidesof

stingso lutionofourproblem~ 3 6 and

cial itleadsdirectand shorttothe

efollowinggeneralproblemofcanal

e the iso lateddistributionofpressure

e ll ingatagi enconstantspeed re uired

splacementofthewater intheplace

before itandbehindit w hichbecomesestablishedafterthemotionof theforci ehasbeenk eptsteady for

resynthesisof thespecialsolution

e so l esnotonly theproblemnow proposed

onfromthe instantoftheapplication

. Thissynthesis thougheasilyputinto

or edouttoanypractica lconclusion. O n

mypresentshort butcompletesolutionof

dymotionfor whichweha ebeen

ngoutil lustrationsin~~ 32-5 .

e finitely asa cur eofsines theD-cur e

2 0 l e a i n g th e f or c i e c u r e F , i s o l at e d

iagrams.O r analyticallystated:



s s_ u s e # p d



R



s s_ u s e # p d



WA V ES



s s_ u s e # p d



R

e u a l v a l ue s o f d ( 0 f o r e u a l po s it i e a n d

f ro m 0~ t o 40 ~ or 5 0~ b y t 8 ) , ( 8 6 , ( 8 7 1

of0ta e

1 8 0~ ) s in ( j + ) 0 .. .. .. .. .. .. 9 1 ,

ca lc ul at ed b y { ( 8 ) , ( 8 6 , ( 8 7 } . T h is u se d in

0 0 f o r al l p os i ti e v a l u e s of 0 g r ea t er t h an 4 0 ~

s i t th e d ou b le o f ( 9 1 f o r al l n eg a ti e v a l ue s o f

.

orPontoons introducedtoapply thegi en

thewater-surface .

amsshowingawater-surface

bef i ed f itt ingclosetothew holew atersurface. Now loo attheforci ecur e F , onthesamediagram

osensiblepressureremo etheco er.

ninsomeparts ofthewholewaterremains

e a m pl e i n f ig s . 1 , 1 4 1 5 1 6 l e t th e

if f co ersf itt ingitto60~ oneachsideof

facebef reef rom60~ to3 00~ ineachof

co ers.Themotionremainsunchanged

dunderthef reeportionsof thesurface. The

egi enforci e andrepresentedby the

isnow automatica llyappliedby theco ers.

8 19 20w ithreferencetothe

heyshow . Thusw eha ethreedifferent

idco er w hichwemayconstructas

ontoon k eptmo ingatastatedv e locity

terbefore it lea esatra inofsinuso ida l

Dcur erepresentsthebottomof the

earrowshowsthedirectionof the

heF cur eshow sthepressureonthe

g.20 thispressureisso smallat-2

supposedtoendthere anditw il l lea e

almoste actlysinusoidaltoan

t( infinitedistanceifthemotionhas

etime . TheF cur eshow sthatin

uidanceasfarbac as-3 q , andinf ig. 18

eepitsinuso idalw henleftf ree q be ingin

a e- length.



s s_ u s e # p d



WA V ES

elessPontoons andthe irF orci es.

chasthoserepresentedinf igs. 18 19

thatifany twoe ualandsim ilarforci es

ance\ X betweencorrespondingpoints and

itutediscausedtotra e latspeede ualto

c co r di n g to ( 7 7 a b o e t h e v e l oc i ty o f f re e wa e s

rw illbe le f tw a eless(atrest behindthe

pletheforci esandspeedsof f igs. 18 19

chforci e inthemannerdef inedin~ 57 w e

sof tw onumbers ta enf romourtables

f igs. 18 19 20 thenumbersw hichgi e

erin thethreecorrespondingwa eless

showngraphicallyinfig.21 onscales

e locity . Thef reew a e- lengthforthis

inthediagram.

andthethreew a elesswater-shapes

show ninf igs. 22 2 , 24ondif ferentscales

pressure chosenforthecon enienceofeach

f thethreecasesta ethatderi ed

na lin estigation. B y loo ingatf ig. 2 w e

ngitsbottomshapedaccordingto the

o + 3 q , 1 f r e e wa e - le n gt h s w i l l le a e t h e

estif itmo esa longthecana latthe

reewa e- lengthis4 . A ndthepressure

mofthepontoonisthat represented

cur e .

scissas ineachofthe fourdiagrams

gedtenfold.Thegreateststeepnessesof

arerenderedsufficientlymoderateto

arealwater-surfaceunderthegi en

b e s ai d o f fi g s. 1 5 1 6 1 8 1 9 2 0 a n d of

habscissasenlargedtw enty fold. Inrespectto

eticsgenerally itisinterestingto remar

retationoftheconditionof infinitesimality

spractica llyallow able . Inclinationstothehori on

( 5 ~ " 7 o r s a y 6 ~ ) , i n a ny r e al c a se o f



s s_ u s e # p d



N W A T ER [ 3 7

~ ~ ~ ~ ~ ~ ~ J ~ ~ ~ ~ ~ ~ ~ r

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ d



s s_ u s e # p d



WA V ES

ances w illnotseriouslyv it ia tethemathematica lresult.

heca lculationsofd 0~ ) and

f o r tw e nt y -n i ne i n te g ra l v a l u e s of j ; 0 1 2 3 , . . . 19 2 0 3 0 4 0 . . . 90 1 0 0 f r o m th e f ol l ow i ng f o rm u la s f o u n d

180~ ; andw ithe= 9ineachcase

+ 1 e [ - e log + 1 + - + + ..

"

1 j ( 2 + 1 e [ l tan- - -- + 1â € ” + +

( - 1 J 2 + 1. .. 9 )

) showninthediagramise plainedby

isinf inite lygreat thetra e ll ingve locityof

small andtherefore byendof~ 41 the

aticallyduetothe forci epressure.

e ualto

f thecur esof f ig. 17forpo ints

dingto integra lva luesof j ise ceedingly

edby it intoanin estigationof the

themotionofasingle forci e e pressed

(94 ;

uturecommunication whenitwillbe

ary toseaship-w a es.

yaidof periodicfunctionsthetwodimensionalship-wa eproblemforinfinitelydeepwater adopted

tion w asgi eninPartIII. ofaseries

a esinF low ingWater publishedin

ine O ctober1886to J anuary1887 w ith



s s_ u s e # p d



S O N W A T ER [ 3 7

forwateroffinitedepths.The annulmentofsinusoidalwa esinfrontof thesourceofdisturbance( a

thecana l by thesuperposit ionofatra in

sw hichdoublethesinuso ida lw a esinthe

December1886 byadiagram[ p. 295supra

wtheresidualdisturbanceof thewaterin

5 a b o e a n d re p re s en t ed i n f ig s . 18 1 9 2 0 .

tothan MrJ . deGraaffHunterfor

ousco-operationw ithmeina llthew or o f

on andforthegreatlabourhehasgi en

andtheirrepresentationbydiagrams.



s s_ u s e # p d



WA V ES



s s_ u s e # p d




~ ~ ~ F



s s_ u s e # p d



eofabscissasisquarter-wa e- lengths.



s s_ u s e # p d



A V ES.

gsof theR oyalSocietyofEdinburgh J uly17 1905

V o l . x i . J a n ua r y 1 9 0 6 p p . 1 â € ” 2 5 .

w emust forthepresent astime

ledinterpretationof thecur esof f ig. 17:

accordingto~ 44 if8= 0( w hichmeans

thedisturbance d isinf inite lygreat o f

ningisclearin( 70 o f~ 3 9.

pressionofthewaterat distancex

thedisturbance isduetoasingle forci e

la

.. . .. . .. . .. . .. . .. . . 9 5 ,

tanyv e locityv . If thisforci ew ere

aceof wateratrestit wouldproducea

) , asw eareta ingthedensityof the

forci e II( x ) w ouldshapethew atertoan

fcross-sectionshowninfig.25 representing

2 o n t he s c al e o f k = 1 0 c m. a n d b = 1 c m.

95 wefindtan- ( x /b .b .Hencethearea

, o r 6 6 . b rb , a n d th e t ot a l ar e a of t h e

f inityoneachside is7rb . Hencethearea

of thetota larea . Thistota larea W rb ,

e f o rc i e a r ea a n d v r b I c a ll t h e me a n

rea. Thebreadthof theforci ew here

n by t h e do t te d l in e B B i n t he d i ag r am i s b .

i n t hi s a nd f o ll o wi n g e p r es s io n s i s t he ( x - t o f

iginofco-ordinatesbe ingnow f i edre lati e ly tothetra e ll ing



s s_ u s e # p d



P S EA S H IP - W A V E S 3 9 5



s s_ u s e # p d



R

besuddenlysetinmotion andk ept

anyve locityv intherightw arddirection

oduce agreatcommotion settling

orenearlysteadymotionthrough

cesfrom 0. Thein estigationof

b. 1904 , andparticularly theresultsdescribed

llustratedinf igs. 2 3 , show thatinourpresentcase

e r v i o l e nt e e n i f in c lu d in g s pl a sh e s , d i i d es

tra elawayinthetwo directionsfrom

-speedincreasinginproportiontos uare

ingtothe law of fa ll ingbodies and

throughe erbroadeningspaces w hatwould

absolutequiescence if theforci ew ere

ingactedforany time longorshort.

ontinuesacting andtra e ll ingrightwardsw ithconstantspeed v , accordingto~ 67 thetra e ll ing

heinitialcommotioninthe two

fmere lyapointo f re ference mo ing

lea esthew ater asshow nby f ig. 26 ina

arlyq uitesteadymotionthroughan

ontherearsideof0 andthroughasmall

pro idedcerta inmoderatingcondit ions

, b v .

e~ 68 f irstsupposev inf initely

nitelylittle disturbedfromthestatic

infig. 25 anddescribedin~ 66. Small

a ev erysmalldisturbancewithany f inite

there tremeandletv bev erygreat.

a lprinciplesw ithoutca lculation thatv

ma ebutv ery littledisturbanceof the

ersteepbethestaticforci ecur e . A

ndaricochettingcannonshot i l lustratethe

namicalprinciplein three-dimensional

hematica lca lculation( ~ 79below w eshall

re a t en o ug h w e ha e

97 ,

dgreatthecommotionis themotionof the li uidis and

onalthroughout.



s s_ u s e # p d



S EA S H IP - W A V E S 3 9



s s_ u s e # p d




htofcrestsabo emeanw ater- le e lin

w a esle f t intherearof thetra e ll ing

e ar e a of t h e fo r ci e c u r e ( f i g. 2 5 ; b e in g

e u a ti o n

. .. . . .. . . 98 :

3 9 ( 7 1 ] by

.. . . .. . . .. 99 ,

hof f reew a estra e ll ingw ithve locityv .

heoreminrespecttoship-wa esis

W i t h ou t c al c ul a ti o n we s e e th a t i f X i s v e r y

w rb( the" meanbreadth o f theforci e

66 , hmustbesimplyproportiona ltoA for

e ll ingatthesamespeed. Thisw esee

a lueofb h/ isthesame andbecause

orci eswithinanybreadthsmallin

esforhthesumof thev a lueswhichthey

artherw ithoutca lculation w ecansee

ca leofourdiagrams thathX / A must

tcalculationIdo notseehowwecould

, asin~ 79below .

tionprescribedin~ 71isillustrated

eringcasesinwhichi tisnotfulfilled.

o forci esbesuperposedw iththeirm iddlesat

g i e h = 0 t h at i s t o sa y n o tr a in o f w a e s .

ceforthis caseisrepresentedinfig.27.

X or-X ; thetw ow illgi ethesamevalue

eonly . Orletthetw obeatdistanceX ;

asgreatasoneforci ema esit.

29 3 0 representingresultso f theca lculationsof~~ 78 79below theabscissasarea llmar edaccording

leofordinatescorresponds ineachof

o k = 2 4 - 89 a nd b rb = 1 02 51 .1 0- . X . T hi s

( 9 7 A = I X , a nd h = t . F i g . 3 0 r ep re se nt s

hema imum intheneighbourhood o f0

e:about1720 timesfortheabscissas

dinates.

theright-handside thew aterslightly

ra e ll ingforci e w hichisadistribution



s s_ u s e # p d



V ES

hosemiddle isatO. O nthe le f tsideof

surfacenotdif feringperceptibly f romacur e

e-lengthrearwardsfrom0.Asmall

hofatruecur eofsinesinthediagram

e ssurfacedif fersf romthecur eofsines

ncef rom0asaq uarter-w a e- length.

atin realitythewatersurfaceis

r ly l e e l a n d in c o ns i de r in g a s w e sh a ll h a e

doneby theforci e w emustinterpret

aggerationofslopesshowninthe

o remar thatthestaticdepression

e ifatrestw ouldproduce isabout87times

roducedabo e0by theforci e tra e ll ing

ew a es o f thew a e- lengthshow ninthe

sinterestinga lsotoremar thatthe lim itationtoverysmallslopesisnotbindingonthestaticforci ecur e .

istributionofstaticpressure e erywhere

urface producingstaticdepression

ig. 25 w ould if causedtotra e lata

w a e- lengthisv ery large incomparison

rbance representedby f ig. 26w ithw a es

assa idin~ 69abo e w ouldproduceno

eedoftra ellingwereinfinitelygreat.

gasshow ingthew a elessdisturbance

andsim ilarforci esw iththeirm iddles

lthew a e- length. Thisdisturbance is

frontandrearof themiddlebetween

namicalconsiderationsof thee uil ibrium

w eseethattheareaof f ig. 27( portion

beingrec onedasnegati e mustbee actly

o f theareasof thetwoforci es representing

wnwardpressure.Thisarea being

er i ca l d at a o f ~ 7 , i s n um e ri c al l y ~ X ; t h at i s

lengthisI andbreadththeunito f

o imatemensuration w ithavery rough

dtherange ofthediagram continued

s v erif iesthisconclusion.

nthesameplanasf ig. 27butw ith

thsasthedistancebetweenthetwoforci es



s s_ u s e # p d



N W A T ER [ 3 8

~ ~ ~ ~ F ;



s s_ u s e # p d



V ES

e- length. Li e f ig. 27 it issymmetrica lon

eof thediagram but insteadofbe ing

27 i t s ho ws f o ur a n d a ha l f wa e s a l l v e r y

al w ithtwodepressiona lha l esofw a es

e le ationscomingasymptotica lly tozero

diagram.Thecur erepresentedby

y theright-hande tremeof f ig. 28: and

rightto le ft isthe le f t-hande tremeof

ththe waterwhollyatrest andstart

erspeed w ithforcegradua lly (orsomew hat

ptotheprescribedamount themotion

resentedby f ig. 28 w ith superimposed

uic lydisappearingine erlengthening

mplitude tra e ll ingaw ay inbothdirections

withtheregularregimerepresentedby

asetoapply theforci es w eha elef ta

a halfv eryappro imatelysinusoidal

ontandarearde iatingfromsinusdidalityas

omtheinstantofbe ingle f tf ree the

itsrearwillrapidlybecomemodified:

central partoftheprocessionwill ha e

lengths w ithvery litt lede iationf romsinuso ida lity. B ut a f terfourorf i eperiodsf romtheinstantofbe ing

processionwillha egotintoconfusion. A f ter

eriods thewaterwillbesensibly

roughthespacewherethe processionwas

partof thespaceo erwhichitwould

ontandrearhadbeenk eptguardedby the

wotra e ll ingforci es. A tnotimeaf ter

escanw ereasonablyorcon eniently

city " to thegrouporprocessionofwa esw ith

. A pre a lentidea is Ibe lie e thatsuch

escouldberegardedastra e ll ingw ithha lf

" o f wa e s o f th e l en g th g i e n i n th e o ri g in a l

e reasonsaregi enforacceptingthetheory

only inthecaseofmutua lly supportinggroups

s S mi t h s P r i e e a m in a ti o n pa p er p u bl i sh e d

ersityC a lendarfor1876: andforre ecting

egroupofw a es. Inreality thef ront

ctseereferencesonp. 304supra .



s s_ u s e # p d



S O N W A TE R [ 3 8



s s_ u s e # p d



V ES

f tra e lsw ithacce leratedve locitye ceeding

w a esof thegi enw a e- length insteadof

teadymotion symmetricalinf rontand

ngforci e w hichisasolutionofour

nstablesolution( asprobablyarethe

of ~ 4 5 a bo e s h ow n i n fi g s. 1 , 1 4 1 5 .

fthewaterisgi eninmotionaccording

a m pl e 5 0 w a e - l e ng t hs p r ec e di n g 0 ( t h e

e- lengthsfo llow ing0 thef ronto f thew hole

of0 w il lbecomedissipatedintononperiodicw a estra e ll ingrightwardsandlef tw ardsw ithincreasing

asingve locit ies andtheappro imately

itw il lshrin bac w ardsrelati e ly to

etheforci ehastra e lledf if tyw a elengths theperiodicw a esinf rontof itarea llgone: butthere

both beforeandbehindit.Afterthe

ahundredwa e- lengths thew holemotionin

emotionforperhaps3 0w a e- lengthsormore

ettledtonearly thecondit ionrepresentedby

sasmallregulare le ationinad anceof

gulartra inofappro imately sinuso ida lwa es

esbeingofdoublethew a e-heightgi en

a s s ai d a bo e i n ~ 6 8 w i ll g o o n l e a i n g

nofsteadyperiodicw a es increasingin

eseanirregulartrainofw a es shorter

dlesshigh thefartherrearwardweloo

0of~ 26 27abo e . It isaninteresting

lem to in estigatethee tremerear

lessw aterbehindit o f thetrainofw a es

etra e ll inguniformly fore er. Ihopeto

henw ecometoconsiderthew or doneby

stigationof theformulasby the

2 6 2 7 2 8 2 9 3 0 h a e b e en d r aw n a n d

spro ed. Gobac totheproblem of

ea d o f ta i n g e= 9 a s i n ~ ~ 4 6 -6 1 t a e

1 / 2 + 1 . B y ( 8 6 a nd ( 8 7 o f~ 4 5 we ha e

100 ,



s s_ u s e # p d



R



s s_ u s e # p d



S H IP - W A V E S 4 05 /

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i



s s_ u s e # p d



R

~ 2 V e s in 10

( j + - 0 ta n- l 2 -e sn ~

e co s 0+ e

g1 - 2 e c. .. 1 01

2 + * e c os j . .. 102 .

2 -3

tedbyputting0= - . andta ing

a ti o n is t h at a s w e sh a ll s e e by ( 7 8 o f ~ 4

( 1 01 , ( 1 02 , e p re ss t he w at er d is tu rb an ce d ue t o an

tconsecuti edistanceseache ua lto

p r es s io n f or e a ch f o rc i e b e in g

n a 2 ~ ~ . .. 1 0 ) ,

. .. . . .. . . .. . . . (1 ) ,

posit i eornegati e integer andby ( 79

. . .. . . .. . . ( 104 .

sureat0dueto eachoftheforci es

o si d es i s 1 / 1 + ( 2 7 r. 1 0 4 2 o f t he p r es s ur e

secentre is0. Thusw eseethatthe

orci es e ceptthe lastmentioned may

era lw a e- lengthsoneachsideof0: and

, ( 1 0 1 , ( 1 0 2 e p r es s t o a v e r y hi g h de g re e

edisturbanceproducedinthewaterby the

ew hosecentre isat0.

a e = 18 0~ i n ( 1 00 , ( 1 01 , ( 1 02 ; w e

e i { V e t an -1 -e 1- e +

. o .

. .( 1 05 .

1-10-4 asw etoo inourca lculations

ow t a e e = 1 . Th i s re d uc e s ( 1 0 5 t o

1 -. .. + 1 j + . .. 1 06 .



s s_ u s e # p d



V ES

te lygreatoddore eninteger andw ef ind

) 7 r .. . .. . .. . .. . .. . .. . 1 0 7 .

a eseen foundbysuperimposingonthe

29aninfinite trainofperiodicwa es

2 7 r / X ) a n d th e re f or e h = r w hi c h

two-dimensionalproblemofcanalship-wa estothethree-dimensionalproblemofsea-ship-wa es we

hodgi enbyR ayle ighattheendofhis

tandingw a esonthesurfaceof running

edtotheLondonMathematicalSocietyin

ninf initeplanee panseofw ater consider

s u ch a s t ha t r ep r es e nt e d by ( 9 5 ' o f ~ 6 6 w i th

eneratinglinesindifferentdirections

a e ll ingw ithuniformvelocity v , inany

onoftheseforci es andofthedisturbancesofthe waterwhichtheyproduce eachcalculatedbyan

( 101 , ( 102 , gi esustheso lutionofathreedimensiona lw a eproblem w hichbecomestheship-w a e-problem

ntsinfinitelysmallandinfinitelynumerous.

nstituentforci easconf inedtoaninfinitely

batedtheconse uenttroublesomeinfinityby

be annulledininterpretationofresults

rto 0.Iescapefromthe troublein

m of w a e s b y t a i n g ( 9 5 t o e p r es s

e intheforci e andma ingbassmall

s i nd i ca t ed i n ~ ~ 7 9 7 , 7 6 b y t a i n g

w e c al c ul a te d a f in i te v a l u e f or d ( 0 . B u t f o r

erablygreaterthanhalfaw a e- length w ew ere

lationsby ta ingb= 0.

nsionalsystemlet inf ig. 31 J be

of therearw ardw a e-normalo foneof

wa es. Thisisalsothe inclination

of thetra e ll ingforci etow hichthat

enow fortheforci eobta inedby the

c. 188 : republishedinRay leigh sScientif icPapers



s s_ u s e # p d



R

numberofconstituents asdescribed

b 2

( x c o s+ y si n) 2 + b 2 * ( 1 08

ionof r andbisthesamefora llva lues

arforci esystemw e

w he re r = 2 y2 .. .. .. 1 09 .

w hethercircularornot bek ept

nofx negati e w ithve locityv : and

pondingf reew a e- lengthgi enby the

thew a e- lengthof theconstituenttra in

gto r= 0. F orthe* -constituent the

rpendiculartothef rontisv cosk , andthe

2. Loo ingnow tof ig. 26 w ithX cos2

othedirectionof themotionof theforci e inf ig. 26.



s s_ u s e # p d



V ES

fg. 3 1 a nd t oe u at io ns ( 9 7 , ( 9 8 ; w e

edepressionat( x , y duetotheconstituentof forci eshow nunderthe integralin( 108 is

x c os r + y s in r . ( 1 1 0

. .

y sinsisconsiderablygreaterthan-X cos2.

t( x , y duetothew holetra e ll ing

s in d 27 r( x c os + + ys in ~ )

, ' k

.. .. .. .. 1 11 .

singthe lim its- - r7r-0 to ris

i egi esatra inofsinuso ida lw a esin

bledisturbanceinits frontatdistances

aw a e- length. Loo now tof ig. 31 and

erofmediallines oftheforci es

108 , ( 111 ; a llaslinespassingthrough

P Y Y L , X X ' o f th es e li ne s ar e sh ow n

dingrespecti e ly to= = - ( 7r-0 ,

e a c ut e a ng l e r = - 7 r. O n e a c h o f th e f ir s t

dicatestherear. Thefourth X X ' , is

on andhasneitherfrontnor rear.

ustincludeall andonlyall themediallines

dsP. HenceQP isone lim ito f rin

ssesthroughP X X ' istheotherlim itbecause

r.Thusallthe linesincludedinthe

seanglePOX ' . Thusthe integral( 111

onatP ( x , y duetothe j o intactionofa ll

becausenonee ceptthosew hosemedial

contributeany thingtothedisturbance

dappro imatelye a luatingthedef inite

enientlyput

= c .. .. .. . 1 1 2 ,

llow s:

rr a

2b 2- si n. .. 1 1 ) .

' X



s s_ u s e # p d



R

erygreat therewillbee ceedingly

ne ua lposit i eandnegati ev a luesof

hichwillcausecance ll ingofa llportionsof the

ifany thereare forw hichdu/ d rv anishes.

ha t t he r e ar e t wo s u ch v a l ue s i * 2 b o th

u b e in g a m a i m u m ( u ) f o r on e o f th e m

fortheother andthat w hen0hasany

a n d 27 r - ta n -~ ^ / t h e v a l u es o f ~ , , 5 2

derationofthislast-mentionedcase

eareaofsea inad anceof two lines

tra e ll ingforci e inclinedate ua l

1 9~ 2 8 ) , o n e ac h s id e o f th e m id - wa e t h er e

ceat distancesofmuchmorethana

hecentreof theforci e . Themain

es therefore l iesintherearw ardangular

ines. It isi llustratedby f ig. 3 2 asw e

y theproperinterpretationof ( 11 ) .

ntof thesinin( 11 ) byTaylor stheorem

ngf rom 1 bysmallf ractionsofaradian

, , . ~ / li /.

- 1 2 = a- ... 114 ,

d 2 u

( ( - ) d- .. . 1 15 .

1 15 w e fi nd d = d l / , 3 i V 7 r , w he re

cttoq2andv a luesof tdifferingbut

+ q ~ 2 i n s t ea d o f th e - q 2 o f ( 1 1 4 , a n d

ad o f th e -( d 2u /d q 2 o f ( 1 1 5 ; b ec au se u i s th e

i ni m um . Ca l li n g k , k t h e v a l ue s o f k

#2 andusingthesee pressionsproperly in

f o r th e d ep r es s io n o f th e wa t er a t ( x , y ,

co s1 d , s i n( a l - q 2

1 17 .

a + q ) . 117 .



s s_ u s e # p d



PSEA SHIP -WA V ES 411

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ c

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~



s s_ u s e # p d



R

assignedtotheintegrationsrelati ely

ca u se t h e gr e at n es s o f r/ X i n ( 1 1 5 a n d th e c or r es p on d in g f or m ul a r el a ti e t o # 2 m a e s q , a n d q 2 e a c h v e r y gr e at

e , fo rmoderateproperly smallposit i eor

- J r a n d - # . N o wa s d is c o e r ed b y

re g or y s E a m pl e s p . 4 79 , w e h a e

2 / 2

w ef ind

- c os a l k , ( s in ~ l + C O S U 2 I

#/ 2c os 2 2 j . .. .. 1 18 .

a luesby (1-15 w ef ind

N

n r ul -

8

l l 8 .

/

ntit iesdenotedby8 , / 2in( 116

e ( 1 1 2 a s fo ll ow s: ru = ( x + y t V i + t 2 w he re t = t a n* . . .. .. .. . 1 1 9 .

ononthesupposit ionofx , y rconstant w e

2l 6. .. .. 1 20 .

y t 5 + t 6 .. .. .( 1 21 .

r t in w h ic h m a e s i t a m a i m um o r m in i mu m

0 .. .. .. .. .. .. .. ( 1 22 ;

wh i ch w h en ( y / X ) 2 & l t - h a s re a l ro o ts a s

x 2

2 4 ~ y + 4 y ~ y I .. . .. . . 1 2 ) .

therof these fort in( 121 , w ef ind

t 2 + 2 yt j \ V 1 + t 1 2. .. .. . 1 2 4 ,



s s_ u s e # p d



V ES

119 ,

x ( 1 + t m 2 3 2 . .. .. .. .. .. . ( 1 2 4 .

rstfactoro f ( 124 by ( 122 w ef ind

( 2 y + 2 t .t mV . .. 1 24 " ,

a nd m = 2 g i e s / a n d / b y ( 1 1 6 .

e s e e th a t ( d 2 u /d 2 M v a n i s he s w he n x = y V 8

o r t a n d po s it i e f or t 2 w he n x & g t y V 8 .

/ dr2negati e . Thereforeu isthema imum

e. Thereforeu2istheminimum and( 119

mumandminimumv alues

V / l+ t 2 r u2 = ( x + y t2 V I + t 2 2. .. ( 1 25 .

2 ) w es ee th at wh en y/ = 0 w eh a e - t = + o o

easey f rom 0to+ x / V 8 - t1fa lls

/ ~ , and- t2risescontinuously f rom

b e co m e e a ch o f t he m V / ~ ; w h ic h

1 6 .

asystem ofautotomic monoparametricco-ordinates . ~~ 87-90.

ru = a . . . . .. . . .. . . .. . . .. . . .. . . .. . 126 ,

ameterO W of thecur eO C C f ig. 3 2

scribe be ingthecur egi enintrinsica lly

w i th s u ff i ' m o m it t ed f r om t . I n th e p re s en t

ybecalledisophasals becausetheargument

be low isthesamefora llpo intsonanyoneof

1 22 f or x a nd y w e fi nd t

t . .. .. .. .. .. . 1 2 7 .

-ordinatesinaplane w eha eaw ell- now ncase inthe

tingofconfocalellipsesandhyperbolas.

p -W a e s ( s u pr a N o . 3 2 p . 3 0 7 s e e Po p . Le c t. I I .

e c h e lo n c ur e s " i s g i e n l i e f i g. 3 2 w i th t h e li n e of c u sp s

meinclination19~ 28 ; butthee uationsaredif ferentf rom

esentingtheef fecto fadif ferenttra e ll ingdistributionofpressure .

i. M ag . V o l . X L V . 1 8 98 p p . 10 6 -1 2 ; L a mb H y dr o dy n am i cs 3 r d e d. 1 9 06 ~ 2 5 , a l so n e wm a tt e r in t h e Ge r ma n t ra n sl a ti o n a n d

c.Roy.Soc. Aug. 1908 assupra p.3 04 whostatesthathis



s s_ u s e # p d



R

r esshowninfig.3 2hasbeen

a luesofx yca lculatedf romthesetw o

n g to - t v a l ue s t an 0 ~ , t a n 10 ~ , t a n 20 ~ , . . .t a n 90 ~ .

alspartiallyshowninfig. 3 2 allsimilar

eendraw ntocorrespondtose ene uidifferentsmallerv a lues 19X , 18X . . . 1 X , o f theparametera ifw e

a l to 2 0 X .

amthate ery tw oof these isophasals

nts ate ua ldistancesonthetw osides

thesystemdow ntoparameter0 e ery

Cistheintersectionof twoandonly

e n b y( 1 2 7 w i th t w o di f fe r en t v a l u es o f t he

completeeachcur ea lgebra ically w e

mbyan e ualandsimilarpatternon

ledpattern thusobta ined w ouldshow

ualandsim ilarinthef rontandrear w hich

p os s ib l e bu t i ns t ab l e. W e a r e h o we e r a t p re s en t

ableship-wa escontainedinthe angle

w o si d es o f t he m i d- w a e a n d we l e a e t h e

honly theremar thata llpointsinthe

m andtheoppositeangle lef tw ardof0

aluesoftheparametera:whileimaginary

real pointsinthetwoobtuseangles.

127 , w ef ind

. .. . . .. . ( 128 ;

istheanglemeasuredanti-cloc w ise

t o th e c ur e a t a ny p o in t ( x , y , i n t he

Eliminationoft betweenthetwo

g i e s a s t he c a rt e si a n e u a ti o n of o u r cu r e

a 2 ( 8 y4 - 2 0y 2 -4 + 6 a 4y 2 = 0. .. 1 29 .

t ions( 127 aremuchmorecon enient

estingtoverif y ( 129 forthecase

2 7 , c o rr e sp o nd i ng t o e it h er o f t he t w o cu s ps

86andthecontinuousvaria tions

ethat- t and- t2arerespecti e ly the

ui a lenttoLordK el in s. Theorigina lreporto f the

pressingthe law ofamplitudeof thew a es w hichare

nt.



s s_ u s e # p d



V ES

ns rec onedf romO Ycloc w ise o f

C andof theshortarcWC intheupper

ifw ecarryapointf rom0toC inthe

W intheshortarc w eha ethechangeof

entedcontinuouslybythedecreaseof

t o 3 5 ~ 1 6 , w hi l e y in c re a se s f ro m 0 t o x / 8

of t a n- l ( - t ) f r om 3 5 ~ 1 6 t o 0 ~ , w hi l e y

0aga in. The inclinationtoO Yof the

t he c u sp C i s 3 5 ~ 1 6 ( o r t an - l V I / .

ortarcC WC of thecur euor

isaminimum. Ineachof the longarcsuisa

po i nt o f t he c u r e t h e v a l ue o f u w h et h er

isa / r. Hencefordifferentpo intsof the

lyproportiona ltotheradiusv ectorf rom0.

18 ' w enow seethatfora llpo intson

r u a n d ru 2 h a e b o th t h e sa m ev a l ue b e in g

thecur e . Thef irstparto f ( 118 ' isone

onatanypoint oneitherofthe long

parto f ( 118 ' isoneconstituentof the

theshortarc. Ta ingfore ample

sshowninf ig. 32 w enow seethatforany

rcs thesecondconstituentofthe

o becalculatedfromthesecondpart of

ranypo into f itsshortarc thesecondconstituent

ca lculatedf romthef irstparto f ( 118 ' .

m ilarly thedeterminationofd( x , y

o f thesmallercur esw hichw eseeinthe

rarcsof the largestcur e w earri eat

sthecompletesolutionof ourproblem.

ingwa esinthewa eofthe

enby thesuperposit ionofconstituents

127 w ithgreaterandsmallerv a luesof

telysmallsuccessi edifferences.

n lo o i n g at t h e wa e s f ro m ab o e i s e a c tl y

andv a lleys w ithridgesandbedsof

cordingtothe isophasa lcur esshownin

yoneof theshortarc- ridgesandfo llowing

w ef inditbecomingthemiddle lineofa

garcsof thecur e . A ndfo llow inga

oughthecusps w ef ind inthecontinuation



s s_ u s e # p d




g ri d ge s . E e r y ri d ge l o ng o r s ho r t i s

lthecur edridgesandv a lleysareparts

o fcur es i l lustratedby f ig. 32and

ra i c e u a ti o n ( 1 2 9 .

nsw emayw rite( 118 ' asfo llow s:

ec si ll ( r u + ) . .. .. 1 0

( + d ) . .. . .. . .. . .. . .. . ( 1 1 .

rhapsthemostimportant featureof the

actua lly seeonthetw osidesof themidw a eofasteamertra e ll ingthroughsmoothw ateratsea oro fa

asfastasitcaninapond isthesteepness

esw hichw ek now tobeinclinedat19~ 28

etheoryof thisfeature ise pressedby

in( 1 0 , andisw elli l lustratedby the

ore le enpo intsofanyoneof the

heresultso fw hichareshowninco lumn6of

ye pressthedepressionbelow and

le e l duetooneconstituentof thesystemof

C o l. 4 C o l. 5 C o l. 6

d 2 a / a s ec 2 t

d \ / ' X

00 100000 1100000 10-000

6 85 ' 9 7 2 9 ' 9 7 8 2 1- 0 64 7

2 0 1 -9 1 58 7 - 7 4 9 7 1- 2 1 0

5 0 ' 8 72 90 3 3 3 3 3 2 - 0 94

3 8 4 9 8 6 6 0 2 0 0 00 0 0 o c

7 7 8 7 22 5 - ' 4 0 8 0 2 - 66 6 0

6 6 - 9 6 2 4 - 1- 8 40 7 0 1 -7 8 9

16 5 1 1 0 94 1 - - 5 00 0 0 1 7 8 88

00 1-5 041 -14-0987 2-279

9 7 2 - 91 2 22 -6 - 3 4 1 4 - 16 7 2

0 0 00 oo - o o

thehighestspeedattainedbyaduc ling thisangle is

terthan19~ 28 becauseof thedynamicef fectof the

ofw ater. SeeB alt imoreLectures p. 59 ( le tterto

2 rdA ug. 1871 andpp. 600 601( lettertoWill iamF roude

6thOct. 1871 .



s s_ u s e # p d



V ES

ysdescribedin~ 92. C o lumn1is-* .

caandy / a ca lculatedf rom( 127 . C o lumn4

2 6 f r om c o lu m ns 2 3 . C o l u mn 5 i s

f rom( 124 ' andco lumns2 4. C o lumn6

f r om ( 1 1 a n d co l um n 1 .u b e in g a s

nm um f o r v a l ue s o f - r fr o m 0 to 3 5 ~ 1 6 , a n d

esf romthisto90~ , w eseethattheproper

forthef irstfourlinesofeachcolumnis2

sis1.

s g e n er a ll y a f un c ti o n of * ; b u t if t h e fo r ci e

e , k i s a c on s ta n t a n d fo r p oi n ts o n o ne o f

a= constant theonlyv ariablecoeff icientsof

d3- l. B utfordif ferentisophasa lcur es

0 e pressingthemagnitudeof therange

e e l v a r ie s i n e r se l y as V / a .F o r m id w a e ( r = 0 a i s s im p ly t h e di s ta n ce f r om t h e fo r ci e : a nd w e

orourpoint- forci e butforagreatship

ery largenumberofw a e- lengthsright

e ightin erse lyasthes uarerooto f the

eorf romthemiddleof theship.

3 5~ 16 representsafeatureana logous

eisin naturenoinfinityforeither

niteanddistributed notinf inite ly intense

ysmall space. Accordingtothe

1-80abo e w eha eine erycaseaf inite

o f fo r ci e e c e pt i n ~ 8 0 wh e re w e h a e

ll incomparisonw ithX , andw ea oid

mn6: andcan bygreatlabour ca lculate

bers risingtoav ery largema imumat

b u t no t t o in f in i ty a n d so a r ri e m at h em a ti c al l y

eryhighw a esseenonthetw obounding

bance inclinedat19~ 28 tothemid-w a e.

memberthatweseein realityaconsiderablenumberofwhite-cappedwa es( would-beinfinities

argeglassyw a esw hichformso interesting

sturbances.



s s_ u s e # p d



S O N W A T E R [ 3 8

tpaperaremerelyaw or ingoutof the

ra itationa lw a esw ithnosurfacetensionontheprinciplegi enbyR ayle igh in188 forthemuch

ofcapil la rywa esinf ront inw hich

efconstituentof theforci e andw a es

hchie fconstituentof theforci e is

ical a lgebra ic graphicof~ 3 2-95

muchva luableassistancef romMrJ . deGraaff

stnow beenappointedtoapostinthe

ory.

c V o l . x v . p p . 69 - 78 1 8 8 ; r e pr i nt e d in L o rd

apers V o l. II. pp. 258-267.



s s_ u s e # p d



DEEP-SEA WA V ESO F THREEC LA SSES:

G LE D IS PL AC EM EN T ( 2 F R O M A G R O U P O F

D I SP L AC EM E NT S ( 3 B Y A P E R I O D I CA L LY

E-PRESSUR E.

. E di n . J a n . 22 1 9 06 P h il . M fa g . V o l . x I I I .

.

toanInitia tiona lF ormmorecon enientthan

r i o us P a pe r s on W a e s . ~ 96 - 11 .

f~ 5- 1 includingthe" f rontand

reeprocessionsofw a esindeepwater

aldisturbances accordingtothe

describedin~ 3 , 4. Inthisform

erywheree le ationore erywhere

mount atgreatdistancesf romtheorigin

es uarerooto f thedistancep f roma

bo ethewater-surfaceinthemiddle

presentpaperanewformof typedisturbanceisderi edindifferentlyfrom eitherthefirstor

sof~~ 3 , 4: f romthef irst bydouble

ncetotime t f romthesecond by

referencetospace x .

f~ ~ 1 2 slightlymodif iedwithrespect

f rictionlessincompressible li uid( ca lled

stra ightcana l inf inite ly longandinfinitely

des. Letitbedisturbedf romitsle e lby

nthesurface uniformine ery line

sides andletitbe lefttoitselfunder

re uiredtofindthe displacement



s s_ u s e # p d



R

particleofwateratany futuretime. Our

yspecif iedbyagi ennormalcomponentofv e locity andagi ennormalcomponentofdisplacement

rface.

tadistance habo etheundisturbed

para lle lto the lengthof thecana l andO Z

etI ~ bethedisplacementcomponents

omponents o fanyparticleof thew ater



ofwateris infinitelysmallincomparisonwiththeirundisturbeddistance andthelinej oining

gesofdirectionwhichareinfinitelysmall

an.W aterbeingassumedincompressibleandfrictionless itsmotion startedprimarilyfromrest

freesurface isessentiallyirrotational.

d

t ; C = d = ; = d... ... ... 1 2 ,

t , o rF asw emayw rite itforbre ityw hencon enient isa functionw hichmaybeca lledthedisplacementpotentia l andF ( x , z , t isw hatiscommonlyca lledthev e locitypotential. Thusak now ledgeof thefunctionF , fo ra llva luesof

pete lydef inesthedisplacementandthev e locityof

edeterminationofF w eha e inv irtue

hefluid

3 ) .

t ion thew ell- now nprimary theoryof

hat if F isgi enfore erypo into f the

andisz eroate erypo intinf inite ly

ofF isdeterminatethroughoutthefluid.

mal andthedensitybeingta enas

fundamenta lhydro ineticsgi es

.

- g + ( z - h~ - € ” ) + . .. 1 4

)



s s_ u s e # p d



A N C E IN I TI A TE D F R O M A G I V E N F O R M 4 2 1

y I1theuniformatmosphericpressureon

p th e p re s su r e at t h e po i nt ( x , z + ' ) w i th i n

x , z , t i s a f un c ti o n wh i ch b e si d es

s a ti s fi e s al s o th e e u a ti o n

5 ;

atthecorrespondingf luidmotionofw hichF

ntia l( 1 2 , hasconstantpressureo er

g ; t h at i s t o sa y e e r y su r fa c e wh i ch w as

sundisturbed. Thusourproblemof

esimalirrotationalmotionofthefluid

sunderanyconstantpressure isso l ed

1 3 ) a nd ( 1 5 .

nw everif y that asfoundin~ 3

- 2 e4 Z + ) . .. .. .. .. .. .. .. .. .. 1 6

nd ( 1 5 . B y c ha ng in g ei nt o - a nd b y

tionsperformedon(1 6 accordingto

d , w h e re i j , k a r e a n y in t eg e rs p o si t i e o r

r e f r om ( 1 6 a n y nu mb e r of i m ag i na r y

ofthese withconstantcoefficients

f realisedsolutions. If asin~ 97 w e

mulasthusobta inedasadisplacementpotential thenby ta ingd/ d o f itw ef ind' thevertica l

t w hichwesha llta easthemost

nineachcaseforthesolutionsw ithw hich

emay ifw eplease ta eanyso lutionof

t ing notadisplacement-potentia l butav e locitypotentia l o rahori ontalcomponentofdisplacementorve locity

tofdisplacementorv elocity.



s s_ u s e # p d



R

12w etoo

+ t ) - 2e4 z ' x )

- os - e ( 1 7 ,

2 , a nd x = t an -l ( x / )

thisnotationband- wasconsistently

w henposit i e upw arddisplacementof

byupw ardordinatesinthedraw ings .

4 f i g. 1 t h at w h ic h h as i t s ma i m um

7 , f o r t= 0 . T h e ot h er c u r e o f f ig . 1

eordinatesonthetwosidesof0

w i th - R D i n st e ad o f { R S . T h e s ym b ol s

e r e i nt r od u ce d i n ~ 3 a b o e { R S t o d en o te

ha lf thesumofw hatisw rittenaf terit

odenotearea li a tionby ta ing1/ 2to f the

us1/2tof thesameformulawith+ t

ur e inw hichtheordinatesare

-o f theordinatesof thesecondof \ / 2LC

isnow gi enintheaccompany ingdiagram

bo eitthef irsto f theo ldcur esof f ig. 1is

atesreducedintheratio2/ 2to1 for

w iththenew cur e. Thisnew cur e

enientinitiationalformreferredtoin

per.

yt a i ng t = 0 i n ( 1 9 o r in ( 1 44 [ m os t

formof ( 1 9 ] , isasfo llow s:

( ' , ' ) 2 = 2 p ( 2 -p ......... 1 8 .

tionofthenewparticularsolution

f r om t h e pr i ma r y ( 1 6 a s i nd i ca t ed i n

efo llow ingformula:



s s_ u s e # p d



A N C E IN I TI A TE D F R O M A G I V E N F O R M 4 2

+ 1 )

{ R D d â € ” ( Z + ) )

g c os _ 4 p2

\ 4p 2 j

2 , a nd X = t a n- ~ ( x / ) . . .. .. .. .( 1 9 .

a forthesamederi ation whichw illbe

n~ ~ 1 5-157below isasfo llow s:

4 z I t ) - ~ ~ __ 0 ( . . .. t

= { R S d t2 â € ” 2 + ( , z t dt .... ... .. ( 1 40 .

1 9 and( 140 iseasilypro edbyremar ing

nd ( 1 5 ,

-gt

= { X ) RS -g z + i - C , + L ... ( 142 .

9 dt \ I z + tX )

3 , andseew ithinhow narrow aspace

+ 2 i n t he n e wc u r e t h e ma i n in i ti a l

w hile intheo ldcur e itspreadssofar

20itamountstoabout-16of thema imum

andaccordingtothe law of in erse

otofdistance w hichholdsforlargev a lues

atx = 80itw ouldstil lbeasmuchas-1of

mparati enarrownessof the init ialdisturbancerepresentedby thenew cur e andtheult imate law of

( insteadofx - 2fortheo ldcur e are

enew cur e intheapplicationsandil lustrationsof thetheory tobegi enin~ ~ 1 5-157below .

tthetotalareaof theo ldcur ef rom

great w hile it iszero forthenew cur e.



s s_ u s e # p d



/ -Tl a. Z d Z - a n oJ ad dy * - c * 1



s s_ u s e # p d




s s_ u s e # p d



R

entialenergyofthe initialdisturbance

2 .. .. .. .. .. .. .. .. ( 1 4 ) ,

odcur e w hile forthenew itisf inite .

maybew ritteninthefo llowingmodif ied

n enientforsomeofourinterpretations

~ 4 p 2

” 2t - CO S c os A. .. .. . 1 4 4 ,

- tan- gt2sinX

X - t an - s . .. .. 1 45 .

- 2 p

w hichforbre ityw eshallca llw atercur esintheaccompany ingsi diagramsof f ig. 3 4 representthe

cordingtoournew solutionr( x , z , t f o r

sp ec ti e ly 0 2 V , , V / wr V 7 r 4 V r 8 V 7 r.

dbyta ingg= 4.Thisismerely

gasourunito f lengthha lf thespacedescended

byabody fall ingf romrestunderthe

orsimplificationinthewriting offormulas

eundisturbedle e lo f thew ater-surface. The

pla inedin~ 107below areca lledargumentcur es asthey representtheargumentof thecosine in( 144 .

ycuriousandv ery interestingfeatureof

reasingnumberofv a luesofx forw hichthe

stimead ances andthe largef igures

u w h ic h i t r ea c he s a t th e t im e s 4 \ / V r a n d

w odiagrams. Thesezeros foranyva lueof t

a ti o n

.. .. . .. . .. . .. .. . .. . 1 4 6 .

highlycomplicatedcharacterofthe

145 thez erosareeasily foundby tracing

w ithA asordinate andx asabscissa( asshown

esof thesi diagramsontwodif ferentscales

tion notformeasurement anddraw ing



s s_ u s e # p d




e atdistancesfromitrepresenting

T e t c. A p a ra l le l a t di s ta n ce - 1 7 i s an

argument-cur es andisshownin

ononescaleofordinates. Thepara llelcorrespondingtodistance-1 - risshowninthef if thandsi th

allersca leofordinatesusedinthe irargumentcur es.

w sz erosatx = + / , o fw hichthat

d1. Intheseconddiagramtheargumentcur e indicatesz erosforthe- rrand- - rparalle ls w hichare

er-cur e . Thezerocorrespondingto

dattheoriginat thetimewhen~ gt2

at i s w he n t w as 1 / V 2 o r ' 7 0 7. I t i s a

sforx -posit i eandx -negati e .

at shortlybefore itst ime ama imum

eintheargument-cur e w hichstil l indicates

aremar edbycrosses.

t inthe inter a lbetweenitstimeand

oz e r os o f t he w a te r -c u r e f o r x - p os i ti e h a e

Theseandthecorrespondingzerosfor

distinctlyonthew ater-cur e andthe ir

earemar edby fourcrossesonthe

betweenitstimeandthatof No.4

a e c o me i n to e i s te n ce o n ea c h si d e of O Z ,

tedfore ampleontheargument-cur e

Nineonlyoutofa llthesi teenzeroson

eonthewater-cur e . These en

oneachside a ll l iebetw eenx= 0and

betweenitstimeandthatof No.5

orx -posit i eha ecomeintoe istence one

by thepara lle l- rr. F ourteenonlyout

rosoneachsideareperceptibleonthe

ofthefiftyimperceptiblez erosoneach

a n d x = + 1 .



s s_ u s e # p d



R

thez erosoriginate inpa irsonthe

x - p os i ti e a n d x - n eg a ti e : t h os e o n th e

intersectionsofoneofthe parallels

1 r/ 2w iththeargument-cur e . The

ment-cur etra e lsslowly intheoutward

1astimead ancesto inf inity . A ttimes

d ia g ra m s 5 an d 6 i t h as r e ac h ed s o c lo s e to

nthasbeenregardedastheactua lposit ionof

orthepurposeofdraw ingthecur e andfor

talnumberof z eros.

ginatesaccordingtoan intersection

eargument-cur etra e lsoutwardsw ith

nfinity astimead ances. Eachof the

ros thatistosay eachzerooriginating

ononthe inw ardsideof theargumentcur e tra e lsveryslow ly inwardsw ithv e locitydim inishingto

estoinfinity.Thusthemotionof the

enx = - 1andx = + 1becomesmoreand

numberof inw ardtra e ll ingw a es

shingtoz ero and asw eseeby the

144 , w ithamplitudesandw ithslopesa lso

ro : astimead ancesto inf inity .

neof theseq uasistandingwa esis

a p pr o i m at e ly e u a l to - 2 p wh e n th e t im e

# gt2isv erygreatincomparisonw ithp.

odisinfiniteat theorigin.Thisagrees

olemotionattheorigin w hich asw e

n ( 1 9 , w it h z = 1 a nd g = 4 i s e p r e ss ed

.. .. . .. . .. . .. . 1 4 7 .

thespacebetweenx = - 1andx = + 1

nterestingcharacter.Towardsafull

ument-cur etothesideof theoriginforx -negati e w e

ev a luesof iin( 146 : butforsimplicityw eha econf ined

positi ev a luesofx .



s s_ u s e # p d




maybecon enienttostudy thesimplif ied

€ ” t 2

- ) ep2. .. . . .. . . . (148 ,

f ( 1 9 gi esw henIgt2isv ery large in

lingz erosonthetw osides beyond

heorigin di idethew aterintoconsecuti eparts ineachofw hichit isw hollye le atedor

w emayca llha lf -w a es. They tra e l

easinglengthandpropagationalv elocity.

d e e l op e d af t er t = / r a s i t tr a e l s

irsttoama imumele ationorma imum

hatdiminishestoz eroastimead ances

tracetheprogressofeachof thez eros

nthetimesofoursi diagrams. Thisis

smar edonse era lo f thez erosinthe

confiningourattentiontotheleft-hand

eindiagram 1asinglez eronumbered1.

e numberedintheorderoftheir coming

, 3 ; 4 4 ... 10 1 0 . .. 3 3 , 3 3 ; . ..a ll in

sdiagram2show sz ero1considerably

thatis outwards ; andz ero2beginningits

m3 showsz eros1and2eachad anced

therthan2. Diagram4show sa llthezeros

istenceattime3 / 7r. Thesearez eros1

w ardsthanattimew / 7r andapa ir 3 , 3 ,

istenceshortlybeforethetime- / Tr.

elsoutwardsandtheinnerinwards.

meintoe istencebetween3 and3 : la ter

istencebetween4and4.

haspassedoutof rangele f tw ards: but

wa r d z e r o s 2 3 , 4 5 6 7 8 9 a n d in d ic a ti o ns o f t he i n wa r d z e r o s 9 8 . T he w ho l e tr a in o f z e r o s f or

eallycontinuedtothe middlebynumbers

5 6 7 8 9 9 8 7 6 5 4 3 ; s i t een in all.



s s_ u s e # p d



R

tof therangeofdiagram6 butw eseein

eros4 5 . . . 12 andanindicationof the

w h ic h h as c o me i n to e i s te n ce b e fo r e th e t im e 8 W / a r.

sfortime8 V /T indicatedbynumbers is

, 3 3 , 3 2 .. .4 3 ; s i t y- fo ur in al l.

the Indef initeE tensionandMultiplicationof

a l Deep-SeaWa esInit ially F inite

7.

standfree afterbeinginitially

nofa finitenumberofsinusoidal

f i emounta insandfourva lleys inthe

eSociety.Theinitialgroup ofwa es

f ig. 35 isformedbyplacingsidebyside

( t a e n a s un i ty , n i ne o f t he c u r e s o f

a lternatelypositi eandnegati e . Diagrams

aremadebycorrespondingsuperposit ionsof

5and6 o f f ig. 3 4. Thusw hat according

ep - se a p er i od i c wa e s ( ~ 1 9 a bo e w o ul d

y thewa e- length if thenumbersof

f inite lygreat w ouldbe2 andasw e

eperiodw ouldbe/ 7r andthepropagationa l

7r.

ewaterisleft free thedisturbance

otwogroupsofw a es seentra e ll ing

themiddle lineofthediagram.The

twogroupse tendrightwardsandleftwardsfromtheendof theinitialsinglestatic group farbeyond

s " supposedtotra e latha lf thew a e elocity w hich(accordingtothedynamicsofOsborneR eyno lds

mportantandinterestingconsiderationof

feedauniformprocessionofw ater-w a es

f thefreegroupsremaineduniform.

grealisedis illustratedbythe

w hichshow agreate tensionoutwardsin eachdirectionfarbeyonddistancestra e lledathalf the" w a e elocity . Whilethere isthisgreate tensionof thef ronts



s s_ u s e # p d



/ M ..

m n . i

J 1 3 5 1 0 1 0 5S

f f i ee le ationsandfourdepressionsemergingastw ogroupstra e ll inginoppositedirections.



s s_ u s e # p d



R

w eseethatthetw ogroups a f ter

stence inthemiddle tra e lw iththeirrears

ebetweenthemofwater notperceptibly

ryminutew a eletsine er-augmenting

ndslowerintherear ofeachgroup.

ereartra elsataspeed closelycorrespondingtothe" halfwa e- elocity " foundbySto esase actly

suniformsuccessionofgroups produced

o-e istentinfiniteprocessionsof

ingslightlydifferentw a e- lengths.

arv elocityisillustratedindiagrams

ndiagram1 R indicatestheperceptible

upcommencingitsrightwardprogress

R showsthepositionreachedattime

s byanidealpo inttra e ll ingrightwardsf rom

eedofha lf thew a e- e locity . This

ndstoafairlywell-mar edperceptible

e ll inggroup.

F , i n t h e t hr e e di a gr a ms o f f ig . 3 5 a n d f f

agram1 F mar saperceptible f ront

ngcomponentgroup.Indiagrams2and

lpo intstra e ll ingrightwardsf romitatspeeds

f wa e - e l oc i ty a n d th e w a e - e l oc i ty . W e

s tu r ba n ce f a r in a d a n ce o f F , F a n d v e r y

w a e-disturbance inf ronto f f f . Thus

elsatspeedactuallyhigherthan the

dthisperceptible f rontbecomesmoreandmore

thewholegroupw iththead anceof time

f ig. 9of~ 20abo e.

bythesediagramshownearly the

cityisfoundin therears:whilethefronts

eaterandwithe er- increasingve locity .

ationsandgraphicalconstructionsof

dtocorrespondingconclusionsinrespecttothe

sion gi eninit iallyasaninfinite lygreat

dalw a estra e ll inginonedirection.

d10 showedrespecti ely attwenty-fi e



s s_ u s e # p d



F A T RA I NO F W A V E S

commencement a f ronte tending

ndaperceptiblerear laggingscarcelytwo

po int tra e ll ingf romtheinit ia lposit ion

half thew a e- e locity .

itia tionandC ontinuedGrow thofaTra inofTwoDimensiona lWa esduetotheSuddenC ommencementofa

V ary ing Surface-P ressure . ~~ 118

gofaf initesinuso idallyvary ing

dk eptthrougha llt imeapplied tothe

a finitepracticallylimitedspaceon

eofthedisturbance.In thebeginning

ereatrestanditssurfacehori onta l. The

f indthee le ationordepressionof thew ater

mid- lineof thew or ingforci e andat

ebegantoact.

119-126 letusconsidertheenergy

fsinusoida lw a es inastra ightcana l

ydeep withv erticalsides.Ifthe

tby anypressureonitsupper surface

lfunderconstanta irpressure w ek now

eticsthatitsmotionwill beirrotational

o lumeof thew ater: andif a tany

esurface isbroughttorest suddenlyor

rate erydepthw illcometorestatthe

faceis broughttorest.This aswe

truee enif the init ia ldisturbance isso

of thew atertobrea aw ay indrops: and

yforeachportionofthewaterdetached

nthecana l asw ellasforthew ater

ifstoppageofsurfacemotionismadefor

before itfa llsbac intothecana l.

onof thew aterisirro tational w eha e

* * * * * * * - ( 1 49 ,

. . .. . ( 149

elocity-potentia l F ha ingbeenta enas

28



s s_ u s e # p d



R

al( ~ 97abo e . A ndbydynamicsfor

s i n ( 6 4 o f ~ 3 8 a b o e

g z - + C . ... ... ... 1 50 .

ondition letz = 1 betheundisturbed

ethevertica lcomponentdisplacementofa

aer ta enpositi ew hendownw ards

tsurface-pressure andta e-asthe

n st a nt C . T hu s ( 1 5 0 g i e s a t t he

, t + g-dt ( , , t + ( 5 1 .

esecondandthirdmembersofthis

rbancebeinginfinitelysmall which

1 + C l t - d F ( x , 1 t a n in fi ni te ly s ma ll q u a nt it y

egligible incomparisonw ithgt1 w hichisan

ofthefirstorder.

e -disturbanceofw a e- length27r/ m

t h v e l oc i ty v , w e h a e a s i n ( 6 6 a b o e

= - k e -n z - l s in m( x - v t ... ... ... 15 2 .

151 becomes

- t - g ( . .. .. .. .. .. .. .. 1 5 ) .

ationofthefreesurface

t . . .. . .. . .. . .. . .. . . 1 5 4 ,

n v / g. . . .. . . .. . . .. . . .. . . .. . . . 155 .

1 52 w it h z = 1 w e fi nd

x - v t . .. . .. . .. . .. . .. . .. 1 5 6 .

154 gi es

2rr.. . . .. . . .. . . .. . . .. . . . 157 .

a c t i i t y t h e ra t e of d o in g w or b y

n onesideuponthewateron theother

x ) . W e h a e

- L g - 1+ ( C ] . .. 15 8 .



s s_ u s e # p d



W T H O F A T RA I NO F W A V E S

n d F b y ( 1 4 9 a n d ( 1 5 2 , w e f in d

v t d e -m z - [ - k m e- m z -1 c o sm ( - v t

.. 1 59 .

operations d , w ef ind

m ( - - v t + g ( - + - ( 160 .

that27r / m istheperiodictimeof the

byW thetota lw or perperiod doneby the

deof theplane( x ) uponthew ateron

h a e

T - = 2 f 2. .. . .. 1 6 1 .

mparethiswiththetotalenergy

+ P perw a e- length. Inthef irstplace

ek ineticenergy K , andthepotentia l

thedensityof thewaterbe ingta enas

2 . .. .. .. .. .. . 1 .6 2 ;

. .. . .. . .. . . 1 6 ) ,

facedisplacement.

52 w e fi nd

c os m ( x - v t . .. .. .. .. .. . 1 6 4 ;

z - 1 s in m ( x - v t . .. .. .. .. .. .. .. ( 1 65 ;

. .. . .. . .. . .. . .. 1 5 6 r e pe a te d .

- -- -2 2 .. .. .. .. k ( 1 66 ;

1 6 6

. . .. . .. . .. . .. . .. . . ( 1 6 7 ,

by (157 .

k ineticenergyperw a e- length

erw a e- length areeache ua ltothe



s s_ u s e # p d



R

by thew ateronthenegati eside uponthe

de o fanyvertica lplaneperpendicular

thecanal. Thusw earri eatthe

now nconclusionthatinaregularprocessionofdeep-seaw a es thew or doneonanyv ertica lplane

yperwa e-length.Thisisonly

ularprocession ad ancingto inf inity

t tra e ll ingw iththew a e- e locityv .

edanideal processionofregularperiodic

ptly tonothingataf ronttra e ll ingw ithha lf

v ; w h ic h i s O s b or n e Re yn o ld s * i m po r ta n t

octrineof " group- e locity .

sionof~ 125isv eryimportantand

two-dimensionalship-wa es.Itshows

regularperiodictra inofw a esintherear

i n e s ti g at e d in ~ ~ 4 8 -5 4 a nd 6 5 -7 9 a bo e

f thespacetra e lledby theforci e f rom

motion butthatitwouldbee actly

difyingpressureweresoappliedto

aras tocausethewa estoremain

ndof thetra in w ithout onthew hole

em orta ingw or f romthem.

ntisapplicabletoour presentsub ect

6 157below .

andf irst insteadofasinuso ida lly

agineappliedaseriesof impulsi epressures

esacertainv elocity-potentialupon

ousimpulses andletitbere uiredto

ity -potentiala tany timet a f tersome

ses.Considerfirstasingle impulseat

say atatimeprecedingthetimetbyan

locity -potentia la tt imet duetothatsingle

liert imet-q bedenotedby

. .. .. .. .. .. .. .. .. .. ( 1 6 8 .

the instantaneouslygeneratedve locitypotentia lisC V ( x , z , 0 , andtheva lueof thisatthebounding

andB rit. A ss. R eport 1877.



s s_ u s e # p d




( x , 1 0 . Hence bye lementaryhydro inetics if I denotesthe impulsi esurface-pressure w eha e

) . . .. . .. . .. . .. . .. . .. . . 1 6 9 .

cessi eimpulsesattimepreceding

q 1 q 2 . .. q i a nd d en ot in g by S ( x , z , t t he

ocity -potentia lsatt imet w ef ind

C1V ( x , z , q , ) + CV ( x , z , q 2 + ... CiV ( x , z , q i ... 1 70 .

estobeat infinitelyshortinter alsof

ormula( 170 intothe languageof the

s: S ( x , z , t = d f ( t - q ) V ( , z , q ) . .. .. .. . ( 1 7 1 ,

no t es a n a rb i tr a ry f u nc t io n o f ( t - q ) a c co r di n g

sure arbitrarilyappliedattime( t-q ) ,

- f t - ) V ( x , 1 0 . .. .. .. .. .. . 1 72 .

dtothe surfaceattimet denotedby

i sa sf ol lo ws :

-f t V ( x , 1 0 ........... 17 ) .

or( 171 gi estheve locity -potentia l

ichfollowsdeterminatelyfromthe

din~ . 127 128. F romit bydif ferentia tionsw ithreferencetox andz andintegrationsw ithrespecttot

mentcomponentsI o fanyparticleof

natesw erex , z w henthef luidw asgi en

em moredirectly andwithconsiderablylesscomplicationofintegralsigns bydirectapplicationof

gasthatusedin( 170 , ( 171 . Thus

z , q ) i n( 1 71 , w es ub st it ut ed V ( x , z , q ) , a nd

z , q ) , w e f in d ~ an d. A nd i f we t a e

q ) and fd d V ( x , z , q ) ...... 174

, q ) i n ( 1 7 1 , w e fi nd t he t wo c om po ne nt s ~ , 4

particle ofthefluid.Confiningour



s s_ u s e # p d



W T H O F A T RA IN O F W A V E S

x , z , 0 , w ou l d be r e ac h ed a n d pa s se d t hr o ug h

gati etopositi e . It isclearthatthe

z , q ) a re e u al f or e u al p os it i e a nd n eg at i e

w he n q = 0 w e ha e

. .. . .. . .. . .. . .. . . 1 8 0 .

h e V ( x , z , q ) , d e f i ne d i n ~ 12 7 w h i c h

0 t o b e an y a rb i tr a ry f u nc t io n o f x , b u t re u i re s

h e n q = 0 s u gg e st s a n al l ie d h yd r o i n et i c

ing(179 w ithW inplaceofV ; and

i n g W = 0 a n d d W / d a n y ar b it r ar y f un c ti o n

siscon enientforourpresentpurpose that

= 0 an dW ( x , z , 0 - 0.. ... ... . 1 81 .

aluesofx andz , largeorsmall

,

0 a nd W ( x , z , q ) 0 .. ... ... . 1 82 .

emtheinit ia tiona lcondit ionis: -displacementz eroandinit ia tiona lve locityv irtua llygi enthroughoutthe

sultof anarbitrarilydistributedimpulsi epressureonthesurface.

iationalconditionis:-thefluidheld

epttoanyarbitrarilyprescribedshapeby

leftfreeby suddenandpermanentannulmentofthispressure.

uestionofacompletesolutionof this

mforanyarbitrary initia tiona ldata w ef indaclass

ntso lutionsinaformulaorigina llygi enin

oya lSocietyofEdinburgh J anuary 1887

g. F ebruary 1887 andusedin~ 3

maynow w ritethatformula inthefollow ing

d e p r es s io n f or V or W : R S o r { R D d e 4 Z + x )

t , w he n i se e n

, w he ni is e o dd en

t , wh en ii sod d



s s_ u s e # p d



R

ma y i n s t ea d o f ( 1 8 ) , t a e t h e fo l lo w in g a s

e : di I ' - t 2

B -d t i ( 1 ) e 4 ( z + x )

t , w he ni is e e n

t , wh en ii sod d

1 7 1 a n d ( 1 7 5 , r e ma r t h at i n te g ra t io n

x , z , q ) = f ) V ( x , z , t - f t V ( x , z , 0

x , z , q ) ... 184 .

adratureorotherwiseweha ecalculated

S ( x , z , t , a s g i e n b y ( 1 7 1 , f o r b ot h f or m s

6 b e lo w w e c an f i nd t h e v e r ti c al c o mp o ne n t

icleof the li uidby ( 175 , w ithout

ormula(184 a lsoshow show bysuccessi e integrationbypartsw ecanreduce

( x , z , q ) . .. .. .. .. .. .. .. 1 85

x , z , t , a s e p r e ss ed i n ( 1 7 1 .

w to ~ 1 2 8 1 2 7 1 1 8: t o m a e t h e a p pl i ed

ary ingpressureput

t - ) . .. .. .. .. .. .. .. 1 86 ;

m a e s

V ( , , 0 . .. .. .. .. .. . 1 8 7 .

fullyw or outourproblemfortw o

nofpressure correspondingtothetwo

, f d e sc r ib e d in ~ ~ 9 6 -1 1 a b o e . F o r t h is

henotationof~ 101

( x , z , t ;

t = ( , z , t = - 0 ( x , z , ... 188 .

a llthesetw ocasescasefandcase~ .

171 and( 175 , e pressingrespecti elythe



s s_ u s e # p d




andthev erticalcomponentdisplacementof

ny time become

= d sin a( t -q ) 0 ( , z , q ) ;

= d cos o ( t- ) ( x , z , q ) . .. 1 89 ; . 0 sin

d | in d ( t- q ) - ( x , z , ) ;

- d s in o( t - q ) ( X , , ) . .. 190 .

f ig s . 3 6 3 7 3 8 a r e ti m e- c ur e s i n

ebeencalculatedbycontinuousq uadrature

ou r f or m ul a s ( 1 8 9 ( 1 9 0 .

. 3 9 b e in g s pa c e cu r e s i n wh i ch t h e

mponentdisplacementsofthewatersurface arethereforepicturesofthewater-surface( greatly

ttoslopesofcourse andmaybeshortly

es.Theirordinatesha ebeencalculated

scribedin~ 151below.Theycannot

y forsuccessi ev a luesofx by the

uadratures ifthatwerethe method

of theordinateforeachv a lueofx w ould

anindependentquadrature( d ) f rom

oftfor whichthewater-surfaceis

e . Theva luesof tchosenforf ig. 39are

+ 1 /8 7 ( i + 2 /8 7 ( i + 3 /8 7 ( i + 4 /8 T whe re

ger andrdenotes27r/o theperiodof the

retowhichthefluidmotionconsidered

a e t a e n c o = V / T r w h ic h m a e s

g = 4 a s in ~ 1 0 5 m a e s t he w a e - le n gt h

7 a l l th e c ur e s c or r es p on d t o

ormulas. Inf ig. 38 a llthecur escorrespondtosinco( t-q ) intheformulas.



s s_ u s e # p d



le.

”



s s_ u s e # p d



le.



s s_ u s e # p d



R

tionsof t imescorrespondtocoso( t-q )

ecur es w iththe inscriptionsa ltered

3 / 8 r ( i + 4/ 8 T ( i + 5 / 8 , ( i + 6 /8 , c o rr es po nd

t he f o rm ul a s.

resentingv e locity -potentia lsandasurface

thecur esshowsanyperceptiblede iation

ptwithinperiod1.Towardstheend of

ndby theq uadraturesshow de iations

hingtoabout1/10percent. andimperceptibleinthedrawings.Thispro esthatsinusoidalityise act

ghall timeaftertheendof thefirst

nperiod1 how nearly therise

ow s t he s a me l a w fo r S ( 0 1 t a n d

standingthev astdif ference inthe law of

representedby ( 188 , forthesetw o

nitiatingsurface-pressurecommences

m a i m um v a l u e - / 2 f or c a se ~ , a n d

theformeris2 8 t imesthe latter.

esubse uentv ariationsofv elocitypotentialshowninthefirstandthird cur esare' 954forcase

ofw hichtheformeris3 ' 00timesthe latter.

a n d fi f th c u r e s o f fi g . 3 7 s h o w a t a

engthf romtheorigin thecompletehistory

dofsurfacedisplacementthroughalltime

cationofpressureto thesurface.The

ccuratesinusoidalityofeachofthesethree

6 7 8 showsthatthecontinuationthrough

sesinusoidal.

withtheinitial agreementbetween

nd S. ( 0 1 t , t ow hi ch we al lu de di n~ 1 9 w ef in d

m ar a b le c o nt r as t b et w ee n S . ( 8 1 t a n d

oughoutthew holeof thef irstperiod. R emembering

ensity thepressure ise ua ltominusthe

ev elocity-potentialperunitoftime

d is p la c em e nt p ( 0 1 t i s a s i s sh o wn

early z erothroughoutthef irstperiod andthat

certainly stil lmorenearly z erothroughoutthef irst

enocur etorepresentit w eseethatthe



s s_ u s e # p d



le.



s s_ u s e # p d




s s_ u s e # p d




t s of t h e sl o pe s i n th e c ur e s f or S ( 8 1 t

r e pr e se n t v e r y ne a rl y t he v a l ue s o f th e a pp l ie d

hew holeof thef irstperiod . Loo now

n e ar t o z e r o i s * ( 8 1 0 , a n d h ow f ar f r om

; a n d w e se e d yn a mi c al l y ho w i t is t h at S . ( 8 1 t

hr o ug h ou t t he f i rs t p er i od a n d S ( 8 1 t i s

andissomew hatneartobe ingsinuso ida l.

ery instructi ecomparisonbetw een

a nd S ( 8 1 t . In th eb ca se f or v a lu es of x a s la rg e

pproachsomew hatnearly tothecaseofa

formsurface-pressureo eraninfiniteplane

therew ouldbenosurfacedisplacement and

he surfacewouldbeate eryinstant

ace-pressureplusthe gra itationalaugmentationofpressurebelowthesurface.Thusweseewhyit is

odicv aria tionofappliedsurface-pressure at

ce lyany riseandfa llo f thesurface le e lthere

alffromthebeginningof themotion

fo r ( 8 1 t .

h andsi thcur esof f ig. 37represent

s es o f d is t ur b an c e S . A S o , a t x = 3 2

theorigin.Ifthe frontofthedisturbance

hew a e- e locity thedisturbancesof the

commencesuddenlyatthe endofperiod4.

1 t a nd ( 3 2 1 t t he d ia gr am s ho ws t ha t

ptibleattheendofperiod4 andbeginto

dofperiod8 w hichw ouldbethee act

wasadef inite" group- e locity " e ua lto

. T he l a rg e ne s s of S o ( 3 2 1 t , a p pr o i m at e ly

rstfourperiods ise pla ineding140.

throughperiods5 6 7 8 dependson

fdisturbancesf romtheorigin asshownfor

t a nd A ( 3 2 1 t i nt he se co nd an df ou rt hc ur e s.

t c ur e o ff ig . 3 8 m ay b ec om pa re d wi th

esignationinfig. 36. Theydif ferbecause

renceinthephase ofcommencementof

whichcommencessuddenlyatits

wa r d or d in a te s i n al l t he c u r e s o f fi g s. 3 6 3 7 3 8 3 9

a luesof theq uantit iesrepresented.



s s_ u s e # p d



R

r esof f ig. 36 andcommencesatzero

3 8 . I f t he S , S c u r e s f or i n it i at i ng

eroweredrawn theywoulddifferfrom

sof f ig. 36inbe ingatthecommencement

bscissas insteadofbeinginclinedto it

asshowninf ig. 36. The ' cur esare

lineofabscissas butthetangency

f ig. 36 w hile it iso f thesecondorder

cur es o f f ig. 38show thew hole

= 0 andx = X , o f thesurfacedisplacement

u la s w h ic h e p s 1 t h e s ur f ac d a s i n c nt d e t o s 1 ) . .. 1 9 1

facedisplacementduetosurface-pressure

in W t ( x , 1 0 . .. .. .. .. .. . 1 9 2 .

8showsthehistory afterperiod3 , to

eriod 9 ofthedisturbanceatthe

isturbancehasnotyetbecomesinuso idal but

moste actlysinusoidalafterafewmore

e ts o f f i e c u r e s s ho w f o r ca s e b an d

alyvary ingw ater-surfaceoneachsideof the

ughtimeafter thebeginningofthe

ularregimeofsinusoida lv ibrationasfaras

thsoneachsideofthe middle.Thethird

ur eofsines. Thef irstcur erepresents

ingofaperiodf romirto( i+ 1 r. The

f irstcur e in erted representsthew atersurfaceatthemiddleof theperiod. Theothertw ocur esmay

tsofthefirst andthird accordingtothe

P s i n wt - Q c o s ac t. .. .. .. .. .. . 1 9 ) ,

cos27r / X . . . . . .. . . .. . . .. . . ( 194 ,

sof thethird fourth andf ifthcur esof f ig. 38isdouble

d indicatedonthefigure.



s s_ u s e # p d




anscendenta lfunctionofx , ha inge ua l

p r es s ed b y ( 1 9 5 f o r po s it i e o r n eg a ti e v a l u e s

w a e - le n gt h .

= - A si n2 7r / . . 1 95

= + A si n2 7r / . .. . " '

i-amplitudeof thev ibration atany time

nning andplacefarenoughfromthe

toha ev eryappro imately sinuso ida l

nofthetranscendentalfunctionQ , and

rbothPandQ , w il lbev irtuallyw or ed

e ceedingly interestingandsuggesti e

cesrepresentedinfig.3 9. Consider

scorrespondingtoPsin cotalone andto

motionPsincot if a tany instantgi en

c o w ou l d co n ti n ue f o r e e r a s a n in f in i te

s withoutanysurface-pressure.Henceour

ssureisonlyre uiredfortheQ -motion:

y instantgi enf romx = -otox = + o

p r o i d ed t h e pr e ss u re - c o s co t ( x , 1 0 i s

dtothesurface.

maybegenera lisedasfo llows: Displacethew ateraccordingtotheformula( 19 w ithPomitted

functionofx formoderatelygreat

a luesofx , gradua lly changingintothe

siti eandnegati ev a luesoutsideany

MON( MO notnecessarilye ua ltoON .

esinusoidallyv aryingsurface-pressure

e uiredtocausethemotiontocontinueaccordingto

uponthemotionthusguidedbysurfacepressure themotion-A cos27r / X . sincot w hichneedsnosurfacepressure . Inthemotionthuscompounded w eha ee ual

e ll ingoutw ardsinthetw odirectionsbeyond

: a n d i n t he s p ac e M N w e ha e a v a r y i ng

perimposingonthemotionPsin cotan

v aryingsinusoidallyaccordingtothe

29



s s_ u s e # p d



R

gdynamicalconsiderationisnow

mponentofmotionneeds asw eha e

ure. TheQ-componentofmotionisk ept

ssureF ( x ) cosot w hich inaperiod

theQ -motion butw or mustbedoneto

otra insofwa estra e ll ingoutw ardsin

thisw or isdoneby theacti ityo f

theP-componentofthemotion.

estionisforceduponus. O ursolution

enusdeterminate lyandunambiguously in

casesconsidered themotionofe eryparticle

espaceoccupied. Thesynthetic

swhichw eha eusedcouldleadtonoother

o theappliedsurface-pressure but

ha e c o ns i de r ed a Q - m o t io n a lo n e k e p t co r re c t

ssure.W ouldthismotionbeunstable

uldit inasuf f iciently longtimesubside into

nthedeterminateso lutionof~ 1 5-145

Atanyinstant sayatt= 0 letthe

omponentaloneof~ 148.Letnowthe

x ) coscot besuddenlycommencedandcontinuedfore eraf ter. Itw il l accordingto~ ~ 1 5-145 produce

ompoundmotion(P Q ) w hichwillbe

otione istingattimet= 0 andthis

gi enw ithitsinf initeamountofenergy

otox = + oo andle f twithnosurfacepressure w ouldclearlyne ercomeappro imate ly toquiescence

ncefrom0 onthetwosides.Thuswe

-motiona loneof~ 148isessentia llyunstable

oesnotsubsideintothe determinate

Itw ouldsosubside if itw eregi en

initespacehow e ergreat oneachside

endistributionofdisturbancethroughany

eatoneachsideof0 le f tto itse lfw ithout

pressure becomesdissipatedawayto

andlea es asil lustratedin~ 96-11 ,

aceoneachsideof0 throughw hichthe

ndsmallerastimead ances.

intosomeoftheanalytical

actical wor ingoutofour solutions



s s_ u s e # p d




T a i n g co s o( t - q ) i n th e fo rm ul as a nd t a i ng c as e

P c o so at + Q s in ot .. .. .. .. .. .. 1 96 ;

z , q ) ; a nd Q d sin w o ( x , z , q ) . .. 197 .

beenthusfoundbyquadratures fora ll

particularv a lueofx , by integrationbyparts

w ereadily f ind w ithoutfartherquadratures

essionsforthese enotherformulasincluded

.

Q fort= oo . U singthe

g i e n by ( 1 7 , w ef in d

c os w~ q e - m ;

d s in w e cm 2 .. .. .. .. .. . ( 1 9 8 ,

+ t X ) .

e a luationgi enbyLaplace in1810*, w e

199 .

isatranscendentfunctionofo andm

ntermsoftrigonometricalfunctionsor

ng t h e se r ie s f or s i n w i n t er m s of ( a ) 2 i + 1

2+ 16m 2by integrationsbyparts w efindthe

riesforthee a luationofQ, fort= oo

1 12 T n 2 . 1. 3 2 r . 1. . V )

m et c

i ~ ~ ~ . .. 2 00 ) ,

S X ( ( t \ I P CO

2 ..

7 + e tc .

O t

a bo e p = V / z 2 + X 2 , an dX = t an- ( x / ) .

i tu t 1 8 10 . S ee G r eg o ry s E a m pl e s p . 4 80 .



s s_ u s e # p d



R

re eryv alueofoV /phowe ergreat.

pgreaterthan4 itdi ergesto large

negati etermsbefore itbeginstocon erge. The largestv a lueofo^ / pforw hichweha eusedit

re sp o nd i ng t o x = 8 a n d re u i ri n g f o r th e

enty-onetermsoftheseries.B utforthis

ra ll la rgerva lues w eha eusedthe

r ie s ( 2 0 8 , f o un d i n e p r es s in g a na l yt i ca l ly n o t me r el y f or t = c a s i n ( 1 9 8 , ( 1 9 9 , ( 2 0 0 , b u t fo r a ll

eatandsmall thegrowthto itsf inal

ofthedisturbanceproducedbyour

licationofpressuretothesurface ofthe

atrest. Thecur eforirinf ig. 39has

by ( 200 forv a luesofx upto8 andby

tseriesforv a luesofx f rom5to10. The

eofthev alueswhichwerecalculated

theult imatelydi ergentseries( 208 ,

oalsowastheagreementbetween

u a d r at u re s f or x = 1 a n d x = 8 w i th v a l ue s

= 1 andby ( 208 forx = 8. It isa lsosatisfactory thattheva luesofP foundbyquadratures forx = 1

we l l wi t h th e ir e a c t v a l u es g i e n b y ( 1 9 9 ,

t t h e e p r e s si o ns ( 1 9 7 f o r P an d Q ,

iousanaly ticalmethodof treatment w e

herefore( ~ 150 a llourotherformulas to

functiondefinedasfollows:

.. . .. . . .. . . .. . 201 ,

mathematicians throughthelast

ndredyears inthemathematical

e f raction andinthetheoryofProbabil it ies. Iha eta enEasanabbre iationofGla isher stnotation

hatheca lls" ErrorF unctionC omplement "

maticaldisco ery de- 2= / 7r seemstoha ebeen

0.

ber1871.



s s_ u s e # p d



R

ergesforallv aluesofa- greatorsmall real

o n e r ge s i n it s f ir s t i te r ms i f 2 c2 & g t 2 i - 3

a-2isimaginary andaf terthatit

a luebeingintermediatebetw eenthesumof

andthissumw iththef irsttermof the

Theproperruleof proceduretofindthe

reeof accuracy istofirst calculateby

tseries andseew hetherornotitgi es

ugh. If itdoesnot usethecon ergent

ch bysuf f iciente penditureofarithmetical

lygi etheresultw ithanydegreeofaccuracy

nly fornumerica lca lculation butfor

o f thedesiredresultw ithoutca lculation it

emodulusesof thethreecomple argumentsof thefunctionE in( 205 , and( 206 . Theyareas

m = / T - s + t - + J

9 I

o c .. .. . . 2 0 9 ;

g ta - 0 2P

= t / -~ + /

\ 4 p p g )

. 2 1 0 ;

.................. 211 .

gq uestionsregardingthefrontof

sine itherdirection o fw hichw eha e

36 3 7 3 8 andw hichw ehadunder

- 1 114-117abo e arenow answ erable

ymathematicalmanner byaidofthe

2 06 , ( 2 09 , ( 2 10 , ( 2 11 . W h en i n th e ar gu me nt s of E i n ( 2 0 5 , a nd ( 2 06 , V / m t is v e r y gr ea t in c om pa ri so n

e tw o a dd e d te r ms i n ( 2 0 5 a r e ap p ro i m at e ly

i s r ed u ce d a pp r o i m at e ly t o i ts l a st t e rm a n d

( 1 9 0 b e co me a p pr o i m at e ly s i nu s oi d al i n

sew hent/ - isv erygreatin



s s_ u s e # p d




andincomparisonw ithw / asw esee

u se s s ho w n in ( 2 0 9 , ( 2 1 0 , ( 2 1 1 . T h is

h e ar g um e nt s o f E in ( 2 0 5 ( 2 0 6 a n d

antre lati e ly tot.

arge andx notsosmallastogi e

ttermsof themoduluses( 209 , ( 210 ,

( 2 06 , ( 1 89 , ( 1 90 a f ul l re pr es en ta ti on o f th e

ew a e- f ront e tendingf romx = obac

thata llow spreponderanceof t/ g

du lu se s ( 2 09 , ( 2 10 . L et f or e a mp le

.

e - e l oc i ty x t i m e. .. . .. 2 1 ) .

efinedis whatinmyfirstpaper tothe

h( J anuary1887 , " O ntheF rontand

on o f W a e s i n De e p W a te r , I c a l le d t he

fn ed i n ( 4 5 o f t ha t p ap e r w h ic h a gr e es w i th o u r

fo llow ingpassagew astheconclusionof that

oaw holly f reeprocessionofw a esmay

aftertheconstitutionof thefronthas

bysuperimposinganannullingsurfacepressureupontheoriginatingpressurerepresentedby ( 12 abo e

) o fourpresentpaper , a f tertheoriginating

edsolong astoproduceaprocessionof

a es. The instructionthusgi enw ith

tweenfrontandrear hasbeenv irtually

somedif ferencesofdeta il in~~ 20-24of

paper onthesamesub ect andunder

20 1904 t. Thatsecondpapercontaineda

calculationsandgraphicrepresentations

thepresentpapercontains inf igs. 35

furtherinstalmentofsuchil lustrations.

, % ~ 9 6 -1 5 7 I h a e h a d mo s t

-t i tl e o nl y .

.



s s_ u s e # p d



R

mMrGeorgeGreen notonly inthev ery

tionsanddraw ings w hichha ebeen

btalsoin manyinterestinganddifficult

dinthefundamentalmathematicsofthe

ongthe methodof~ 128tocalculate

( x - v q , z , t - q ) , t he i ni ti at io n an d

lShip-w a es duetothesudden

nuedapplicationofamo ing steady

, 1 0 . W e h o pe a l so t o a pp l y ( 1 9 o f t he

mentofmyoldpromise( ~ 3 0 J une20

lwiththebeautifullyv ary ingprocessionseen

placeof astonethrownintodeep

ev ariousgraphs ca lculatedinthesepapersasrepresenting

pesofdisturbance ha ebeenanalysedandv erif iedin

fromthepointo fv iew ofgroupv elocityorLordKel in s

ase( supra p. 304 , intw opapers" O nGroupV elocity

fWa esinaDispersi eMedium " P roc. R . S. Edin.

p p . 44 5 -4 7 0 a n d " O n W a e s i n a Di s pe r si e M e di u m re s ul t in g

trbance " P roc. R . S. Edin. V o l. x x x . 1909 pp. 1-12.



s s_ u s e # p d



4 57 )

T IO N O T H E MA C E R EL S Y .

cia tionR eport 1876 P t. II. p. 54 reprintedfromSymons s

ag a i n e V o l . x i . 1 87 6 p . 1 1 .

pla inedthere lationof thecloudsand

andthatitw asnotessentialto theformationof

hereshouldbetw odifferenttemperatures.

thatportionsofairshouldbe mo ing

er thattheupanddownmotionshould

from theslippingofonestratumof air

ductiontherebyofw a es andthesecond

her ofthetwoportionsof airshould

saturation-thatitwouldbeclearwhen

ndcloudywhenupatitshighest .

" UeberatmospharischeB ew egungen ( 1888 andfo llow ing

Papers V o l. III. p. 289se .



s s_ u s e # p d



[ 4 1

MATIC A LA NDDYNA MIC A LTHEOR EMS.

gsof theR oyalSocietyofEdinburgh V o l. v .

11 â € ” 1 15 .

ationswhichtheauthorhadbeenled

tha TreatiseonNaturalPhilosophy

tare abouttopublish hemetwith

ems w hichappeartobenew andof

Asthedetailsofthe in estigationswill

e rybrief s etchonly isgi enhere .

htwire ofuniformsection

rencetracedonitssurfacepara lle lto its

diculartothislinef romanypo into f the

erse theamountof torsionortwisto f the

ny form maybedeterminedby thefo llowing

hetangenttothea iso f thew ire atapo int

radiusofanunitspherebedraw n cutting

cur e . F rompointsof thiscur edraw

sesatthecorrespondingpointsofthe bar.

eof directionfromonepointtothe other

the increaseof itsinclinationtothetrans erse

hecorrespondingparto f thew ire .

curiousconse uencesfo llow o fw hichone

benta longanycur eonaspherica l

neofreferenceliesall alongincontact

T ai t s l Y at u ra l P hi l os o ph y ~ 1 2 .



s s_ u s e # p d



C A LA NDDYNA MIC A LTHEOR EMS

uiresnotw ist sothatw henanapple

peeled there isnotw istinthepeel.

rrowribbandbe laidonasurface

stw istisate erypo inte ua ltothe

alsystematrest andsub ectedtoan

agnitudeandinanyspecif ieddirection it

a ethegreatestamountofk ineticenergy

secangi e it.

po intsbestruc independentlyby

inamount morek ineticenergy isgenerated

freetomo eeachindependentlyofall

connectedinanyway.

asystematrest. Letanypartso f it

yw ithgi env e locit ies theotherparts

eirconnectionswiththose whichare

esystemw illmo esoastoha eless

ngstoany othermotionfulfillingthe

ons.

Ta it sNat. Phil. ~ 311.

Ta it sNat. Phil. ~ 3 12.



s s_ u s e # p d



R M O F C EN T RI U G A L G O V E R NO R .

gineersinScotland Transactions V o l. x II. No . 25

ora centrifugalgo ernoristouse the

eproducedbyincreaseof.speed without

stheforcetoproducethere uisiteregulating

ayof usingthisforceforthe purpose

pressure forafrictionalarrangement

gtherotatorymotion.

ntotheInstitution isofthisperfectly

presentsnono eltye ceptinsomedeta ils

portionofits parts.Itconsistsoftwo

MM(seeP late III. ) , eachsuspendedf roma

H attachedtotheshaf t S andturningw ith

chisv ertical. Thesemassesarepre ented

alforceby astoutringof gunmetal

diameter f i edhori ontally a taboutthe

nertia . B utthegreaterparto f the

cedbypow erfulsprings P drawingthe

th e a i s . F i r m s to p s F , a r e pl a ce d l e e l

ia topre entthembeingdraw ninwards

ninchfromthe positionwhichthey

egun metalring.W henthemachine

asingve locity thego erningmassesdo

suntilthecentrifugalforce uponthem

theforceof thesprings. A v erysmall

o ethatw hichf irstdetachesthemfrom

pressagainstthegunmetalring and

sistanceimpedingfurtheraugmentation

ringsofeachmassareatav eryconsiderabledistanceapart( 5inchesinthe instrumente hibitedto

re .



s s_ u s e # p d



O R MO F C EN TR I U G A L GO V E RN O R

planeperpendiculartothehori onta ll ine

tothea is. Thisgi esgreaterf irmness

suspendedmass v erynearlyasif itw ere

ta lshaf t butw ithoutthefrict ionw hich

ouldentail.I tallowsthehori ontal

ntheleadmasswithoutsensibly

andtobetransmittedto therigid

esist itsmotion.Thespringwhichdraws

is ismadeupof twopiecesofstout

edandtemperedproperly placedw iththeir

soneanother andpressedagainstoneanother

her andunitingthembystoutclamps.

adaptedforpullinginsteadofpushing.

the Institutioneachmassamountsto

resetbyanad ustingscrew sothat

otherbe ingtiedinbyacord beginsto

eandthesamespeedis reached.This

minute inthe instrumentasad usted

ution and therefore asthecentreof

out41inchesf romthea is itscentrifugal

tsw eight or48poundsw eight w hich

orcew ithw hichthespringw asad ustedto

eedisincreasedbyasmallpercentageabo e

hego erningmasstobegintopressupon

w hicheachwillpresswille ceedtheforce

esamepercentage.Thus ifthespeed

hatatw hichthego ernorbeginstoact

heringw ithaforceof ' 19ofapound

tional resistanceof.0ofapoundforce

be' 105.Thusthewholefrictional

masseswillbea-of apoundactingat

tf romthea is andconsuming therefore

nd.Toincreasethe speedfurtherby

somuchincreaseofdri ingpowerasto

rsecondmore.These figuresgi ea

rof thisgo ernorwhenusedsimplyto

itionalwor donebyadditionstothe

orethanasmall increaseofspeed.The

e thatthegreatestadmissiblepercentage

gi efrictionalresistanceamountingto



s s_ u s e # p d



ermittedchangeofdri ingpower and

ngpowerspentinf rict iononthepi ots

inproportiontothe latter. Itw as

formiscellaneouslaboratorypurposes

inwhichappro imatelyuniformspeed

anmaybeuseful forchronoscopesin

capparatus whetherforgi inguniform

ribbonofpaper asintheMorseandother

anica l" sending instruments.

wsaplan in entedbyProfessor

andintroducedbyhiminconnectionw ith

a lgo ernor tobeappliedtothepresent

ertedintoapow erfulsteamgo ernor.

thegunmetalring andsupportingitso

rotateroundthesamevertica la isasthe

ts. B yanycon enientmechanism a

samedirectionasthatof thego ernor

ndrotationinthecontrarydirectionaugment

sed andaspringorw eightappliedto

tterdirectionw henit isnotcarried

of thego erningmasses. Thus the

the Institutiongi esthemeansof

persecondofwor toactincuttingof f

espeedaugmentsbyonepercent.

inesa idthatthisw asago ernorof

indeed asSirWm. Thomsonhadsa id itw as

stofall principlesthatcouldbeapplied

ethere o l ingmassespressaga instthe

c edthespeedw henitbecametoo

attheprincipleonwhichthe go ernor

hadnotpre iouslybeenappliedinpractice .

efficacyinpreser inganalmostuniform

tthat itwouldbepracticableto adapt

oughnotprecise ly initspresentform but

dificationsit couldbeadaptedtothat

ethe meansofregulatingthespeed

recisionw hichtheyhadheardstated and

dif ferentia lgo ernor.



s s_ u s e # p d



O R MO F C EN TR I U G A L GO V E RN O R

dsee beforethemeetingbro eup.

attheUni ersity inSirWill iam

oratory ofseeingcontri ancesofthis

hattheyga eresultsastouniformityof

theyhadhearddescribedinthe paper.

chtheysaw onthetable thego ernor

nfriction.Itwaseasy tounderstand

onsitmightbe madetoactupona

ecut-of fo fasteamengine.

thatwiththeappliance towhich

couldberegulatedw ithv erygreatprecision forinstance easily soastok eepthespeedw ithinaha lf

mity.Inreferenceto whatProfessor

hepossibil ityo f thisgo ernorbe ing

es ago ernorgoingatdoubletheve locity

notbesensiblydisturbedby thero ll ingof the

osedthatinusing thisasasteam

rn g wo u ld b e r e u i re d f or t h e re o l i n g

henthespeed fellshort inordertoact

onthe regulator.

therewerespringsand stopswhich

ay. HemightdoeitherwhatDrR an ine

ghtbeaf rict iona lactionof thego ernorto

wlydescendingweightalwaysthrowingon

thef rict iona lactionba lancesoro ercomesit.

e effectedbyawheelcarriedround

oughta weightwouldbethemost

chwouldberunningdow nuntilthefull

beforefullsteamwasadmittedthespeed

ausethemassestopress againstthering

weightfromrunningdownany further

oesteamfromgettingin.

R N O M I CA L C LO C , A N D A P EN D U L U M

R U N I O R M M O T IO N .

v I I . J u n e 1 0 1 8 69 p p 4 6 8- 4 70 . Re p ri n te d i n

resses i i . pp. 387- 94.



s s_ u s e # p d



A T IO N S O T H E CO M P A S S PR O D U C E D

THESHIP.

heB rit ishA ssociationatB e lfast 1874 f romthe

V o l . X L V I I I . N o . 1 8 74 p p . 3 6 - 6 9 .

whichhasbeenin estigatedbyAiry

isthede iationof thecompassproduced

asaconstantinclinationof theshiprounda

pro imatelyhori onta l iscalled . Itdepends

entof theship smagneticforce introduced

ch compoundedw iththehori ontalcomponente istingw hentheshipisupright gi esthea ltered

hentheshipis inclined.Regardingonly

anddisregardingthechangeoftheintensity

wemaydefinetheheeling-errorasthe

ns fortheshipuprightand forthe

sultanto f thehori ontalmagneticforces

sitionofthecompass.Thesesuppositionswouldbe rigorouslyreali edwiththecompasssupported

manner ifthebearing-pointwere

mlyinastraightline. Theyarenearly

geshiptorenderinconsiderablethe

ectuniformityofthemotionofthe

o intisplacedanywhere inthe" a iso f

eshipthecompass how e erplaced isnot

ypitching orby the ine ua litiesof the

otioncausedbyw a es. Hence supposingthecompassplacedinthea iso f ro ll ing theperturbation

gwillbe solelythatduetothe

eofA rchiba ldSmith P roc. Roy . Soc. 1874 tobereprintedina laterv o lume a lsoanarticle inPopularLecturesandA ddresses V o l. III.

h ichformedthebeginningsofcompassin estigations.

y thebestinpractice o f f indingbyobser ationtheposit ion

ohangpendulumsf rompointsatdif ferentle e lsinthe

perpendiculartothedec til lone isfoundw hichindicates

as thosefoundgeometricallybyobser ingagraduated

) seenaga instthehori on.



s s_ u s e # p d



I O N S O F T H E CO M P A S S B Y R O L L I NG

nta lcomponentof theship smagnetic

hecompasswould-ha eonegreat

icationofpropermagneticcorrectors

awayw iththero ll ing-error w ould

g-error.Tosetoff againstthis

wopractica ldisad antages: -one thatthe

a lw aysbelow dec w ouldnotbeacon enientposit ionfortheordinarymodesofusingthecompass

ous , that ata lle entsinshipsw ithiron

ticdisturbanceproducedby the ironof theship

chgreateratanypo into f thea iso f

ychosenposit ionsabo edec , astomore

randk ineticad antageof thea ia l

sin shipsofv ariousclassesoughtto

found thatinsomecasesthe compass

tngad antage beplacedatthea iso f

er thisposit ionforthecompasshasnot

class and asw eha eseen it isnot

rbegenerallyadoptedforshipsof all

nterestingandimportantpractical

perturbationsofthecompassproduced

-uniformmotionsofthe bearingpoint.

lemofthecompassis todetermine

ofarigidbodyconsisting ofthe

andf ly -card w hichforbre ityw illbecalled

o ableonabearing-po int w henthispo int

nmotion. Letthebearing-po inte perience

cce lerationa inanygi endirection.

rw eight o f thecompass andgW the

rec onedink ineticunits. Theposit ion

of thecompassatthatinstantisthe

restunderthemagneticforcesand

itye ua ltotheresultanto fgW anda

oppositeto thatofa.Nowtheweight

tanditscentreofgra ity so low that

carcelyaffectedsensiblybythe greatest

encedby theneedles . Hence ink inetic

ustingcounterpo iseforthecompassisre uiredw henaship

thtoe tremesouthmagneticla titudes.

3 0



s s_ u s e # p d



hecompass-cardis sensiblyperpendicular

apparentgra ity def inedabo e and

eneedlesisinthedirectionof the

nts inthisplane o f themagneticforces

issimplythroughtheapparent

theshipoccupiedby thecompass dif fering

n- le e l thattheproblemof thek inetice uil ibriumposit ionof thecompassinaro ll ingshipdiffersf rom

errorreferredto abo e.Thatwemay

iesofour presentproblem letthere

heshipherse lforcargo. Thek inetice uil ibriumposit ionof themagnetica iso f thecompassw illbe

ponentofterrestrialmagneticforcein

tle e l. LetK bethe inclinationof this

ra itation- le e l andb& gt thea imuth

f rommagneticnorthof the lineLL o f

planes(adiagramisunnecessary ;

hori onta landv erticalcomponentsof

rce.Thecomponentofthis forcein

e lw il lbetheresultanto fHcos/ a long

Z s i n K p e rp e nd i cu l ar t o L L ; a n d th e re f or e i f & g t , d e no t e th e a ng l e at w h ic h i t i s in c li n ed t o L L , w e ha e

n K Z s i n Ic

t a n ~ c o s IC + r - .

Ho

uestions w erec onthedirectionsasof

oles( orthenorthernendsof the

hedirectionofHcosbisa longLL northw ards

nK, w hentheshipisanyw herenorthof

isdow nw ardsintheplaneof theapparent

nsideringtheef fecto f ro ll ing the

acce leration o f thebearing-po intwill

endiculartotheship slength and

aa lle lto the length. ( Itw il l infactbe

bber-points ofthecompass-bowl.

anglesareordinarilyread intheplane

etice uil ibrium-erroro f thecompassis

b . W h e n fc i s a s ma l l fr a ct i on o f 5 7~ ' 3

stheanglew hosearcise ua ltoradiushas

amesThomson w hichisthecase



s s_ u s e # p d




egreesof ro ll ingwhenthecompassisproperly

p pr o i m at e ly

s forthenorthernendsofthe needles

emisphereorfor thesouthernends

e towardsthesideon whichtheapparent

is( aspractica lly thecompassisa lways

ing , tow ardsthee le atedsideof theship. It

e

tosay w hentheshipheadsnorthorsouth

mount considerperfectlyregular

ra lfulf i lsappro imate ly thesimpleharmoniclaw sothatw emayput

nationof theshipattimet andnand

heheightofthebearing-pointof the

abo ethea iso f ro ll ingw hentheshipis

ontof itsacce lerationw eha e

n 2h i.

gthofasimplependulumisochronous

p w eha e

-gh/ l. i .

gtangentialtothecircle describedby

pro imate lyhori ontal andthereforethe

ityw illbeappro imately thato f the

ri ontal.

. i a p pr o i m at e ly .

eadsnorthorsouth theamountof the

rrorisappro imate ly

ompass perniciouslyusedintoomanymerchantsteamers

ghro lling e periencede iationsofapparentle e lamounting

chsideof thetruegra itation- le e l.



s s_ u s e # p d



le theperiodof theroll ingtobe6seconds

odof the" seconds pendulum ) ; 1w ill

esthe lengthof theseconds pendulum .

stobe14- feetabo ethea iso f ro ll ing.

sothattherangeofapparentro ll ingindicated

pointinthepositionof thebearingpointofthecompassisgreater byhalfthanthetrue rangeof

ppositionsthek inetic-e uil ibriumerror

rit ishIslandsthemagneticdipis70~ ,

e ingthenaturaltangentof thedip ise ua l

hek inetic-e uil ibrium errorforthe

thislocalityto aboutadegreeand

degreeof ro ll.

ibriumvalueof thero ll ing-errorw ill

umof thek ineticerrorin estigatedabo e

byanin estigationreadilywor ed

SmithintheAdmiraltyCompass

SectionIV . pages82-89 andA ppendi ,

thmodificationtota e intoaccountthe

ntle e l a ttheplaceof thecompass f rom

el.

s si o n " k i n et i c- e u i li b ri u m e r r or t o

estigatedabo ef romthatactua lly

ss. It ise actly theerrorw hichw ould

passwithinfinitelyshortperiodof

c needle( e itherw ithsil - f ibresuspension

theordinaryw ay ha ingaperiodof

conds showstherolling-errorv ery

te ery instanta lmoste actly theposition

. Iha ethusfoundthero llingand

asmallwoodensailing- esselthatit

oma ee actobser ationsw iththequic

eF rithofC lydeoroutatseaontheA tlantic

e ceptiona lly smooth. Thew ell- now n

eforashipofanysi ee posedtoregularw a esof

stocrest and ifmo ingthroughthew ater mo ingina

crests.



s s_ u s e # p d




cedoscil la t ions isreadilyappliedtoca lculate

anironship theactua l" ro ll ing-error

he" k inetic-e uil ibriumerror in estigated

ecompassatany instant f romthe

theshipw ereatrestandupright

scillationifunresistedby any

ce( thedampingef fecto fcopper introducedbySnow Harrisandusedwithgoodef fectinthe

ass beingincludedinthis

theamountof v iscousresistance

briumvalueof thero ll ing-error

g.

2 r /T a n d n = 2 7 r/ T . T h e di f fe r en t ia l

s

2 E c os n t .

toe presstheeffectofregular

4n 2f -2 '

2 n f

entcommunicationtoofarto enter

orthepresentit isenoughtosay

ofv iscousresistancecanma ethe

forpracticalcon enience unlessalso

s longerthanthatofany considerablerollingtowhichtheship maybesub ected.Probablya

econds( suchasanordinarycompass

cessary forgeneraluseatsea andit

cticalq uestionhowisthisbest tobe

thesmallnessofthecompass-needles

satisfactoryapplicationofthesystem

whichAiry proposedtocausethe

pointcorrectmagneticcourseson all



s s_ u s e # p d



R M O F A S r RO N O M I CA L CL O C W I T HF R E E

P EN D EN T LY G O V E RN E D U N I O R M M O T I O N

HEEL.

rNo. 42( 1869 supra PopularLecturesandA ddresses

isreproduced w iththefo llowingaddit ion. F rom

876 P t. i i . pp. 49-52.

hopeheree pressedhasnothitherto

earpassedproducingonlymore orless

ousmechanica ldeta ilso f thego ernorand

untilaboutsi monthsago w hen forthe

le ceptthependulumsinappro imately satisfactorycondit ion. B y thattimeIhaddisco eredthatmycho ice

he temperaturecompensationandlead

ulumswasamista e.Ihadfallen into

oughbeinginformedthatinRussiathe

enre ertedtobecauseofthe difficulty

mperaturethroughoutthelengthofthe

tstoppingtoperce i ethattherightw ay

wastofaceit andta emeansofsecuring

peraturethroughoutthelengthofthe

ob iousmaybedonebysimpleenough

isedapenduluminwhichthecompensationis

fz incandaplatinumwire placednearly

throughoutthelengthofthependulum

f thecloc showntotheB rit ishA ssociation

an.Nowit isclearthatthe materials

should o fa llthosenototherw iseob ectionable bethoseofgreatestandof leaste pansibil ity . Therefore

numoughttobeoneof thematerials and

stronomicalmercurypendulumisa

ttobetheother( itscubice pansion

are pansionofz inc , unlessthecapil la ry



s s_ u s e # p d



F A S T RO N O M I CA L C LO C

surfaceleadto irregularchangesin

Theweightofthependulumoughtto

testspecif icgra ityattainable ata ll

e istobemountedinanair- t ightcase

rors ofthebeste istingpendulumsis

ariationsofbarometricpressure.The

tsitouto f theq uestionforthew eightof

ena lthoughtheuseofmercury forthetemperaturecompensationdidnota lsogi emercury forthew eight.

odcompensationcouldbegot byz incand

means mercuryought onaccountof its

y tobepreferredto leadfortheweightof

madese era lpendulums( fort idegauges w ithnoothermateria linthemo ingpartthanglassand

dk nife -edgesofagateforthef i edsupport

ma ingfourmorefortwonew cloc s

eontheplanw hichformsthesub ectof

ehadnoopportunityhithertoof testing

hesependulums buttheiractionseems

esults andtheonlyuntowardcircumstance

edinconne ionwiththemhasbeen

intwoattemptstoha eonecarriedsafely

madeby MrW hitetoanorder forthe

new cloc , it isenoughto loo atthe

perfectsteadiness frommonthtomonth

entimetreoneachsideof itsmiddle

tsonly touchedduring3 -0of thetimeby

tofee lcerta inthat if thebestordinary

sanyofitsirregularitiesto v ariationsof

orto impulsesandfrict ionof itsescapementw heel thenew cloc must w hentriedw ithane ua llygood

reregular. Ihopesoontoha eittriedw ith

atof anyastronomicalcloc hitherto

wsirregularitiesamountingto-iofthose

oc s thene tstepmustbeto inclose it

tatconstanttemperature dayandnight



s s_ u s e # p d



D A S PO S S I B L Y A M O D E O M O T I O N .

n Pr o c. V o l . I . M ar c h 4 1 8 81 p p . 52 0 5 2 1 P o pu l ar

V o l. I. pp. 142-146 orpp. 149-15 inla terreprint.

t leo fhisdiscoursethespea ersa id:

ynda ll sbeautifulboo , Heat aModeof

uthwhichhasmanifestedfarand wide

e greatestdisco eriesofmodern

ysadmiredit Iha e longco etedit

b y k i n d p er m is s io n o f it s i n e n to r I h a e

ning sdiscourse .

goDanielB ernoullishadowedforth

elasticity ofgases whichhasbeenaccepted

endidlyde e lopedbyC lausiusandMa w ell

swayingsofacrowdtoobser ation

eepathof anindi idualatominTait

ationofC roo es granddisco eryof theradiometer andinthev i idrea lisationof theo ldLucretiantorrents

mselfhasfollow edupthe ire planationof

ments byw hich lessthantw ohundred

erybyR obertB oy le ' theSpringofA ir

estatisticalresultantofmyriadsof

atomsmustha eelasticity andthis

nedbymotionbeforetheuncertainsound

of thediscourse ' E lasticityv iew edas

canbera isedtothegloriouscerta inty

M ot i on ' .



s s_ u s e # p d



O F M O T I O N

spinning-tops thechild srollinghoop

otionascasesofstif f e lastic- l i e

t ion andshow ede perimentsw ith

positions utterlyunstablewithout

nedwithafirmnessand strengthand

bybandsofsteel.A fle ibleendless

ausedtorun rapidlyroundapulley and

off thepulley andletfallto thef loor stood

itsmotionwaslost byimpactand

ef loor. A limpdiscof indiarubber

emedtoac uirethestiffnessofa

Alittlewoodenball whichwhen

erj umpedupagainina moment

ddedinj ellywhenthewaterwascaused

prangbac , asif thew aterhadelasticity

henitw asstruc byastif fw irepusheddow n

cor byw hichtheglassv esse lconta ining

stly la rgesmo eringsdischargedf rom

ure inabo w ererenderedv isible by

intheirprogressthroughtheair ofthe

ular anditsmotionwassteadywhen

proceededwas circular andwhenit

erring.W henoneringwassent

hecollisionor approachtocollisionsent

ngeddirections andeachv ibrating

bberband.W hentheaperturewas

ngwasseen tobeina stateofregular

nning andtocontinuesothroughoutits

room. Here then inw ateranda irw as

o lid de e lopedbymeremotion. May

ryultimateatom ofmatterbethus

sk inetictheoryofmatterisadream andcan

tcane pla inchemicala f f inity e lectricity

ion andthe inertiao fmasses( thatis crowds

tgi eane planationofgra ityandof

asses onthevorte theory w ere itnot

pyofcrystals andtheseeminglyperfect

70 .



s s_ u s e # p d



DYNA MIC S [ 46 47

nger-postpointingtowardsawaythat

mountingofthisdifficulty oraturning

ndisco ered orimaginedasdisco erable .

yof matterispossibleisthe onlyground

s instoreforthe worldanother

ledElasticity aModeofMotion.

A KINETIC THEOR YOF MA TTER .

ciationR eport Montrea l 1884 pp. 61 -622 P residentia l

ntedin PopularLecturesandAddresses

rpp. 225-229inlaterreprint.



s s_ u s e # p d



W O R I N G MO D E L O F T H E

ciationR eport 1884 pp. 625-628.

eB ritishAssociationatSouthport ,

methodsforo ercomingthediff icultiesw hich

Ibe lie e a llpre iousattemptstorea lise

deaofdisco eringwithperfectdefiniteness

motionbymeansof thegyroscope. Oneof

llymyselfput inpracticewithpartially

sa

easuringtheV erticalComponentof

yrostatssupportedonk nifeedges

ase withtheirlineperpendicularto

lyw heelandabo ethecentreofgra ity

w or byane ceedingly smallhe ight

he ldw iththea iso f thef lywheelandthe

hori ontal andthek nifeedgesdow nwards

rmingtheir function.Theapparatus

nifeedgeswiththeflywheelnotspinning

eamofanordinarybalance.Let now

smallk nifeedges ork nife -edgedho les

ofanordinaryba lance gi ingbearing

cuttingtheline ofthek nifeedgesas

dofcourse( unlessthere isreasontothe

municationhas so farasIk now hithertoappearedin

eport188 , p. 405 gi esthetit le ' Gy rostaticDetermination

neandtheLatitudeofanyP lace. ]



s s_ u s e # p d



hef ramewor appro imatelyperpendiculartothisline and forcon enienceofputtingonandoff

anordinarybalance tw overy lightpansby

ntheusua lw ay . Now w iththef lyw heel

byw eightsinthepansifnecessary sothat

e uil ibriuminacerta inmar edposition

ninclinedslightly tothehori onta lin

ef lyw heel w hetherspinningoratrest

s topressononeand notontheother

ongingto itstwoends. Now unhoo the

gy rostatandspinit replace itonits

nthetw opans andf indthew eightre uired

edpositionwiththeflywheelnowrotating

byanob iousformulaw hichw asplaced

hport gi esanaccuratemeasureof the

heearth srotation* .

pingNeedle.

rtthatthegy rostaticba lancedescribedabo e ifmodif iedby f i ingthek nifeedgesw iththe ir

s possiblethroughthecentreof

ndf ramew or andw iththefacesof the

ttheysha llperformthe irfunctionproperly

yw heelispara lle lto theearth sa iso f

nofthe flywheelinthesamedirectionas

ustasdoesanordinarymagneticdipping

titude insteadofdip anddippingthe

ow nw ardsinsteadof theendthatis

he magneticdippingneedle.Thus

eedgesbeplacedEastandWest the

itsa ispara lle lto theearth sa is and

outhend downwardsinnorthern

downwardsinsouthernlatitudes.

on andlefttoitself itwilloscillate

amelaw asthatbywhichthe magnetic

a-1TWk 2oysinI w herew denotestheba lancingw eight

uponit a thearmonw hichthisforceacts J V thew eight

adiusofgy ration o itsangularve locity y theearth s

dIthe latitudeof theplace.



s s_ u s e # p d



O R I N G MO D E L O F M A GN E TI C C O M P AS S 4 77

roundina imuththepositionof

esamelawas doesthatofa magnetic

ealtw ith. Thus if the lineofk nife

thegyrostatwillbalancewiththe

rtica l andifdisplacedf romthisposit ion

gtothesamelaw butw ithdirecti e

of the latitude intothedirecti ecouple

ineofk nifeedgesisEastandWest. Thus

esusthemeansof definitelymeasuring

srotation andtheangularve locityof

Ibelie e bev eryeasilyperformed

elfhithertofoundtimeto trythem.

eticConmpass .

agyrostatsupportedfrictionlessly

i s w i th t h e a i s o f th e f ly w he e l ho r i o n ta l

ustasdoesthemagneticcompass butw ith

mcalNorth ( thatistosay rotationa lNorth

orth. Ialsoshowedamethodof mounting

e itf reetoturnroundatrulyv erticala is

tionalinfluenceasnotto pre entthe

emethod how e er promisedtobe

andIha esincefoundthattheob ectof

delofthe magneticcompassmay with

ynamica lmodif ication bemuchmoresimply

dingthegyrostatbya v erylongfine

t withsufficientstabilityonaproperly

stigatethetheoryofthisarrangement

statw iththea iso f itsf lywheel

gbyav ery f inewireattachedto itsf ramewor atapo int asfarascancon enientlybearrangedfor abo e

f lyw heelandf ramew or andlettheupper

edtoatorsionhead capableofbeing

erticala isasinaC oulomb storsion

mplicity le tussupposetheearthtobenot

ngsetintorapidrotation letthe

gementsha ebeeninrecenttimestriede perimenta lly in

na ies andarenow inactualuse.



s s_ u s e # p d



w ire andaf terbe ingsteadiedascarefullyaspossiblebyhand letitbe le f tto itse lf . If itbeobser ed

imutha lly ine itherdirection chec this

ad thatistosay turnthetorsionhead

ositetotheobser eda imutha lmotion

endo nothingtothetorsionhead

rsea imutha lmotionsuper enes. If itdoes

yopposingitby torsion butmoregently

en thetorsionheadisleft untouched

t.Theprocess gonethroughwillha e

omwhatwouldha ehadtobe performed

tatw ithitsrotatingflyw heel arigidbody

utwithmuchgreatermomentofinertia

s hadbeeninitsplace. Theformulafor

finertiaisas follows.Denoteby

dedw eightof f lyw heelandf ramew or ,

rationroundthevertica lthroughthe

wholemassregardedfora

wheel

onof thef lyw heel

po into fattachmentof thewireabo e

f lyw heelandf ramew or ,

yonunitmass

ityof thef lyw heel thev irtualmoment

a la isis

a . . .. . .. . .. . .. . .. . .. .

Hereitis. Denoteby

f i edv ertica lplaneandthev ertical

so f thef lyw heelatany timet

posedtobeinfinitely smallandintheplane

isinclinedtothevertica lat

e t or u e r ou n d th e v e r t i ca l a i s e e r te d

suspendedflywheeland



s s_ u s e # p d



O R I N G MO D E L O F M A GN E TI C C O M P AS S 4 7 9

of momentofmomentumroundan

ea iso f rotationre uisitetoturnthe

angularv e locitydp/ dt w eha e

. .. . . .. . . .. . . .. . . . 2 ,

momentof thecouple inthevertica lplane

hthe angularmotiondo/dtinthehori ontalplaneisproduced. Againbythesameprinciple of

momentumta eninconnectionwith

faccelerationofangularv elocity we

- F . . . .. . .. . .. . .. . .. . . 3 ) .

see uationswefind

.. . .. . . .. ( 4 ,

ctionofHingeneratinga imutha lmotion

f asinglerigidbodyof momentof

m ul a ( 1 a s s ai d a bo e w e re s u bs t it u te d

icmodelcompass:arrangea

recedingdescriptionwitha v eryfine

tlessthan5or10metreslong( the longer

ufficientlyshelteredenclosurecon eniently

senforthee periment . P roceedprecise ly

rostattorestbya idof thetorsionhead

rooforothercon enientsupportsharing

on.Supposefora momentthelocalityof

thertheNorthorSouthpole theoperation

hegyrostattorestwill notbedisco erablydifferentfromwhatitwas aswefirstimaginedit whenthe

ot rotating.Theonlydifferencewill

ostathangsatrestre lati e ly totheearth

allconstantva lue sosmallthatthe

rtica lw il lbequite imperceptible unless

inglysmallthatthearrangementshould

sco erw hichw astheob ectof thegyrostatic

abo e thatistosay todisco erthe



s s_ u s e # p d



heearth srotation. Inrealityw eha e

n eniently can anditsinclinationto

rebev erysmall w henthemomentof the

ahori onta la isperpendiculartothe

lyw heelisj ustsuf ficienttocausethea is

ndwiththeearth.

ywheree ceptattheNorthor

insteadofbringingthegyrostattorestat

bringittorestbysuccessi etria lsina

udgingby thetorsionheadandtheposit ion

ethatthere isnotorsionof thew ire . In

thegy rostatwillbe intheNorthand

e uil ibriumbeingstable thedirectionof

stbethe sameasthatofthe componentrotationoftheearthroundtheNorth andSouthhori ontal

sacasetobea o idedinpractice thetorsional

eatastocon ertintostabil ity the

erotorsionalrigidity therotational

inrespecttothee uil ibriumof the

ersedf romtheposit ionofgy rostatic

r ed how e er thate enthoughthe

eatthat thereweretwostable

theposit ionofgyrostaticunstablee uil ibriummadestableby torsionw ouldnotbethatarri edat: the

ce uilibrium renderedmorestableby

osit ionarri edat by thenatura lprocess

alwaysinthedirectionoffindingby

uilibriumwiththewireuntwistedby

nhead.

orsionheadbringthe gyrostatinto

isinclined atanyangleb tothatposit ion

suntwisted itwillbe foundthatthe

aance it inanyobli ueposit ionw illbe

ngthis descriptionresultsfrom

irtualmomentofinertia represented

bo e. Thepaperatpresentcommunicated

aculationsonthissub ect w hichthrow

caldifficultieshithertofeltin any



s s_ u s e # p d



O R I N G MO D E L O F M A GN E TI C C O M P AS S 4 81

y rostaticin estigationof theearth s

e ledtheauthorto fa llbac uponthe

tSouthport ofwhichtheessential

nthe frameofthegyrostatinsuch a

ustonedegreeof f reedomtomo e. The

escriptionofa simplifiedmannerof

gyrostaticcompass-thatistosay

naplaneeitherrigorouslyorv ery

nta l.



s s_ u s e # p d



ERIMENTS.

theB elfastNaturalHistoryandPhilosophical

89 pp. 89-91( A bstract . . . . THER Eare how e er otherproblemsconnectedw ithgy rostaticswhicharefarmorediff iculttosol e . The lecturerne t

bilityofdifferentformsof gyrostats

e andordinarydiscand gimbal-formed.

gso lutionofC olumbus sproblemhow to

nd. If theeggishard-bo iledit is

ndli eatop w hereasthev iscousf luid

sitsbeingtreatedinasim ilarmanner.

ee ertoso l ethedif ficultproblemof

itwillbeby theaidofthe phenomenaof

dulatorytheoryoflight weshallsee

antdisco erydemonstratesthegy rostatic

uttheinfluencedueto rotationcould

asF aradaydisco ered isproducedby

singthroughglass betweenthepolesof

gowewereall tryingtofindsomek ind

ev ibrationsoflightandelectricity and

ebyProfessorF it Gerald( w homhewas

sent fouryearsagoatSouthportga ethe

eq uestion.Hesuggestedtheemployment

andthatsuggestionhadbeenrealisedin

w ithinthepastyear andthegapw hichw e

filled.Itis almostimpossibletogoa

sanddynamicswithouttheaid of

he reasonthelecturerisinterestedin

thephenomenatheypresentarecurious



s s_ u s e # p d



MENTS

l es. B utinstudy ingandreconcil ing

wsofmagnetism the lawsofe lectricity

ityofmatter gyrostaticsplayanundoubtedlyimportantpart.

hanumberof e perimentstending

staticdominationingi ingstability

etc. Oneof themostinterestinge amples

o ew reathsorrings demonstratingthe

nsodelicate amedium. Another

stheimpartingofstability towaterby

otion.Inconclusion hesaidthat

chhehadendea ouredtogi esome

anscomplete yetitw illdoubtlessintime

eanw hileany thingthattendstoad ance

desired endisworthyofour attention.



s s_ u s e # p d



AS E S F O R T HE M A X W E L L - B O L T Z M A N N

DISTRIB U TIONO F ENERGY .

. V o l . L . J u n e 11 1 8 91 p p . 79 - 88 N a tu r e

- 5 8.

s a rt i cl e ( P h il . M ag . 1 86 0 " O n t h e

s " enunciatesavery remar able

portanceinthek inetictheoryofgases

assemblageoflargenumbersofmutually

o f se era ldifferentmagnitudes the

thesamefore ua lnumbersof thespheres

assesanddiameters or inotherwords

es uaresof thev e locit iesof indi idua l

sthe irmasses. Themathematica lin estigationgi enasaproofof thistheoreminthatfirstart icleonthe

tisfactory butthemereenunciationof it

w asav eryva luablecontributiontoscience.

" Dynamica lTheoryofGases " Phil. T rans.

w el l f in d s in h i s e u a ti o n ( 3 4 ( C ol l ec t ed W o r s

athoroughmathematica lin estigation the

dto includeco ll isionsbetw eenB osco ich

accordingtoany law ofdistance pro idedonly thatnotmorethantw opo intsare inco llision(thatis

ncesof the irmutua linf luence simul [ Inadiscussionensuing.onthispaper theposit ionofB o lt mannandMa w ellw assupported amongothers byLordR ay le igh Phil. Mag. V o l. x x x IIi . 1892

Papers V o l. II. pp. 554-7 cf . a lsoPhil. Mag. V o l. X LIX .

c ie n ti f ic P a pe r s V o l . i . p p . 4 3 - 4 51 . T he s u b e c t i s d i sc u ss e d

ordKel ininthesecondparto f theR oya lInstitution

nthC enturyC loudso ertheDynamicalTheoryofLightand

d F e b r ua r y 2 1 9 01 r e pr i nt e d as A p pe n di B o f B a l t im or e

527.



s s_ u s e # p d



O L T Z M A NN P A RT I TI O N O F E N ER G Y

a well soriginaltheoremforcolliding

tudesinaninterestingand important

b e c t in ~ ~ 1 9 2 0 2 1 o f hi s p ap e r " O n t h e

neticTheoryofGases" ( Trans. R . S. E. for

StudientiberdasGle ichgewichtder

schenbew egtenmateriellenPun ten ( Sit b.

c t o b er 8 1 8 68 , e n un c ia t ed a l a rg e e t e ns i on o f

w ellasti llw idergenera lisationinhispaper

sTheoremontheA erageDistributionofEnergy

o ints ( C ambridgePhil. Soc. Trans.

shedinv o l. II. o fMa w ell sScientif icPapers

hefo llow ingef fect( p. 716 : Intheult imatestateof thesystem thea eragek inetic

rt ionsof thesystemmustbe intheratioof

eedomof thoseportions.

asbeen feltastothe complete

ofcasesforw hichthere istruth o f this

dif feringaslitt leaspossible f romMa w ell s

spheres considerahollowspherical

obuleweshallcallitfor bre itywithintheshell.Imustfirst digresstoremar thatwhathas

Clausiusandothersbeforeandafter

ityan" e lasticsphere " isnotane lastic

nandofelasticdeformation andtherefore

berof modesofsteadyv ibration into

edegreesofnoda lsub-di isionandshorter

translationa lenergyw ould if theB o lt mannMa w ellgenera lisedpropositionw eretrue beult imate ly transformedbyco ll isions. The" smoothe lasticspheres arerea lly

s w iththeirtranslationa linertia andw ith

forceate erydistancebetw eentw opo ints

eradiiof thetw oballs andinfinite

distance. WemayuseB osco ichsim ilarly fortheho llow shellw ithglobule initsinterior andsodo

stov ibrationsduetoelasticityof material

heglobule.Letus simplysupposethe

eshellandthe globuletobenothing



s s_ u s e # p d



coll ision andthentobesuchthatthe ir

locitya longtheradiusthroughthepo into f

heco ll ision w hilethemotionof the ir

nchanged.

sha llca lltheshellandinteriorglobuleof

ecule orsometimes formorebre ity adoublet.

ere" of~ 3 willbecalledsimplyan atom

theradiusordiameterorsurfaceof the

sordiameterorsurfaceof thecorrespondingsphere. ( Thise planationisnecessary toa o idan

curwithreferencetothe common

faction o faB osco ichatom.

umberofatomsanddoublets

df i edsurface ha ingtheproperty

componentve locityofapproachofany

ttheinstantof contactofsurfaces

edtheabso lutev e locityof thecentreof

yve locityorve locit iesinanydirection

oanyoneormoreof theatomsorof the

ngthedoublets. Accordingtothe

ldoctrine themotionw illbecomedistributed

thatult imate ly thetime-a eragek inetic

achshell andeachglobulesha llbee ua l

doubletdouble thatofeachatom.

mar ellousconclusion butIseeno reason

t. A ftera ll it isnotob iouslymore

minglyw ellpro edconclusion thatina

o ll id ingsingleatoms someofwhichha ea

assof others thesmallermasseswill

il iontimestheve locityof the larger. B ut

w ell sproof forsingleatomsofdifferent

his" DynamicalTheoryofGases re ferredto

condit ionthattheglobulesenclosedintheshells

ellsfromcollisions withoneanother

to n [ ( C o f ~ 1 8 o f " F o u nd a ti o ns o f K . T .

tthere isperfectly f reeaccessforco ll isionbetween

herof thesameorofdifferentsystems.

gationofsuchasimpleand definitecase

oubletsdef inedin~ ~ 3 -5isdesirable



s s_ u s e # p d



M A N N P AR TI T IO N O E N ER GY

terestingasan illustrationweretestnot

ceedinglyw idegenera lisationsetforthinthe

ldoctrine .

onlyasingleglobulew ithintheshe llo f

astnumber. Tof i ideasletthemassof the

dredtimesthesumof themassesof the

mberof theglobulesbeahundred million

beconnectedbyapush-and-pull

egi enatrest w iththespring

t andthenle f tf ree . A ccordingtothe

ldoctrine themotionproducedinit ia llyby

ributedthroughthe system sothat

ineticenergiesoftheglobuleswithin

dmillionmilliontimesthe a erage

ell. Thea eragev elocity o f theshell

d-millionthofthea eragev elocityof

ingpropositioninthe k inetictheory

rigidshe llseachw eighing1gram and

monatomicgas beattachedtothetwo

fectlye lastictuningfor andsetto

ilbecomeheatedinv irtueof itsv iscousresistancetothev ibratione citedinitby thev ibrationof theshe ll

nergyofthetuningfor isthusspent.

ublemoleculesof~ 5 supposethe

onnectedbymasslessspringswiththe

gedtowardsthecentreofthe shellwith

tothe distancebetweenthecentresofthe

w hichIga e inmyB alt imoreLectures

ionforv ibratorymoleculesembeddedin

a lenttotw omassesconnectedbyamassless

motionsinonelinetoconsider butithas

perfectly isotropic andgi ingfora ll

edlinee actly thesameresultasif

endiculartoit. When apairofmasses

i eaf i edobstacleoramo ablebody

esnote actlyperpendiculartothe

l o c it y o f a pa r ti c le " i r re s pe c ti e l y of d i re c ti o n i s ( i n t he

acon eniente pressionforthes uarerooto f thetimes

eof itsv e locity .



s s_ u s e # p d



it iscausedtorotate. Nosuchcomplication affectsourisotropicdoublet. An assemblageofsuch

mo ing aboutw ithin a rigid enclosing

aestatist icsbe foreachdoublet , e ua l

iesofmotionofcentreof inertia andof

o constituents

uestionsynthetically w ef inda

meprobleminthedetailsof allbutthe

o ll isionw hichcanoccur w hichisdirect

pre iouslyv ibratingdoublets orany

ouslyv ibratingdoubletagainsta f i ed

emassesofglobuleandshellaree ua l

tsoftwo impactsataninter aloftime

of f reev ibrationof thedoublet andaf ter

sseparationw ithoutv ibration j ustasif

sinsteadof thedoublets. B utinobli ue

pre iouslyv ibratingdoublets e enif

obulearee ua l w eha easomewhat

ndtheinter albetweenthetwoimpacts

eragek ineticenergiesof thetwoconstituents and con erse ly e ua la eragek ineticenergiesof thetw oconstituents e ceptinthecaseof

a l impliesthee ua lity statedinthete t. Letu u beabsolutecomponentv e locit iesof twomasses m m , perpendiculartoaf i edplane

omponentv e locityof the ircentreof inertia andrthato f

otion. Weha e

U + , - . .. .. . .. . .. . .. . .. . . 1 ; ? - t I . .. .. . .. . .. . ..

4 mm

2 = ( i ) - n . + U r . .. .. . .. . .. . .. . . ( 2

erageofU rtobez ero . Ine erycase inwhichthisis

,

u 2 ( in-m ) x Time-a .U 2( - + m ) J ...... 3 ) .

h

me-a .m u 2........................... 4

) x T im e- a . U 2 â € ” ( , I + f ) = 0 . . .. .. .. .. .. .. .. .. .. . 5 ,

pt w h en m 1 i= i , w e m us t h a e

) U 2= Time-a . .....................( 6 ,

si t io n b e ca u se a s w e re a di l y se e f ro m ( 1 , t m me 2 / m + i m )

h e k i n e t ic e n er g y of t h e re l at i e m o ti o ns u - U , a n d U - u .



s s_ u s e # p d




andtof indthef ina lresult ingv ibration. When

motionparalleltothe tangentplaneofthe

certa inva luedependingontheradiusof

he ll theperiodof f reev ibrationof the

i ev e locityofapproach there isnosecond

oubletsseparatew ithnorelati ev e locityperpendiculartothetangentplane buteachwiththeenergyof that

usmotioncon ertedintov ibrationa lenergy.

ll ismuchsmallerthanthe massofthe

te eryco llisionwillconsisto fa large

mse ceedinglydifficulttofindhowto

f thesechatteringcoll isions andarri eat

ultimatedistributionofenergyinany

esotherthanMa w ell sorigina lcaseof

t mann-Ma w ellgenera liseddoctrine istrue

itstruthasessentia l w ithspecia l

ases e enw ithoutgo ingthroughthe

hedetails.I canfindnothingin

cleonthesub ect( C amb. Phil. T rans. May6

hs p re i o us p a pe r s p r o i n g an a f fi r ma t i e

of~ 7.

le ttheglobulesbe init ia llydistributed

ogeneouslythroughthehollow leteach

neighboursbymasslesssprings andlet

neartheinnersurfaceof theshellbe

masslesssprings.O rletanynumberof

withinour outershell andconnectedby

entedbytheaccompanyingdiagram

myB altimoreLecturesnow inprogress.

gi enatrestw iththe irsystemsof

w ithinthem beconnectedbymassless

dinmotion asw eretheshe llso f~ 6. There

ossofenergyfrom

swhich therewasin

theult imatea erage

oletwohundred

becertainly small

matea eragek ineticenergyofthesingleshell. Itmaybe

6isfree towanderthattheenergy



s s_ u s e # p d



tcase anddistributedamongthem.

ntheirmotionallowingthemto ta e

nowwhentheyareconnectedbythe

posethemotionsinfinitesimal orif

aybe allforcesarein simpleproportiontodisplacements theelementarydynamicaltheoremof

show tofinddeterminatelyeachof

si simpleharmonicv ibrationsof

fromtheprescribed initialcircumstancesisconstituted.Ittells usthatthesum ofthepotential

achmoderemainsalwaysof constant

me-a erageof thechangingk ineticenergy

hisconstantv alue. Withoutfully

the600mill ionmill ionandsi coordinates it iseasy toseethatthegra estfundamenta lmodeof

cedinthe prescribedcircumstances

ndenergyfromthesinglesimple

hthetw oshellsw ouldta e if theglobules

them orw ereremo edf romwithin

itia lcircumstancesw erethoseof~ 6. B ut

theforces beingrigorouslyinsimple

ts.

ldtheybeso andif there isany

eproportionalityofforceto displacement

sitionofmotionsdoesnotholdgood.

mof fundamenta lmodes a lthough sofar

yhasnotyetbeenin estigated- . F orany

w ithagi ensum E o fpotentia land

eremustingenera lbeatleastasmany

orouslyperiodicmotionasthereare

entv ariables . B uttheconfigurationof

snownotgenerallysimilar:for different

erpositionofdifferent.fundamentalmodes

rw ithdif ferentva luesofE hasnow no

obablethat e eryfundamentalmode

edJ uly10 1891.

nc a re M e ca n i u e C6 e es t e o r a s q u o te d i nf r a p . 5 11 .

miccases thatistosay casesinwhichthere isno

fore ample aparticleconstra inedtoremainonasurfaceand

clineunderthe influenceofno" applied force.



s s_ u s e # p d



M A N N P AR TI T IO N O E N ER GY

sso ifMa w ell sfundamental

hesystemif lef tto itse lf in. itsactua lstate

nerorla ter passthroughe eryphasew hich

uationofenergy istrue. Itseemsto

thisassumptionistrue pro idedthe

isnote actly astoposit ionandv elocity

eofthe fundamentalmodesofrigorously

ro ideda lsothatthe" system hasnotany

suchasthose indicatedbyMa w ellfor

usthathisassumptiondoesnot hold

a w ell sfundamenta lassumption Ido

ca lw or ingsofhispaper+ anyproofo f

ea eragek ineticenergycorrespondingto

sisthesamefore eryoneof thev ariables

asagenera lproposit ionitsmeaningis

eemstomeine plicable . Thereductionof

umofs uares~ lea esthese era lparts

spondencetoanydefinedordefinable

ables. What fore ample canthemeaning

hecaseofa j o intedpendulum ( asystem

esupportedonaf i ed hori onta la isand

isf i edre lati e ly tothef irstbody

gra ity . Theconclusionisq uite

b u t is i t t r ue ) w he n t he k i n et i c en e rg y

mofs uaresof ratesofchangeofsingle

iedbyafunctionofa ll o ro f some of

e fore ample thestilleasiercaseof

t.

arlytheeasiestcaseof all motion

ne thatisthecaseof j usttw o

s a y x , y a n d k i n e ti c e ne r gy e u a l to

u a t i on s o f mo t io n a re

dV

y

lenergy w hichmaybeany functionof

o l. II. p. 714. + Ibid. pp. 714 715.

~ Ibid. p. 722.

" b " i np .7 2 .



s s_ u s e # p d



l y to t h e co n di t io n ( r e u i re d f or s t ab i li t y t h at i t

i t s l ea s t v a l u e be i ng f o r br e i t y t a e n a s

e d t ha t w i t h a ny g i e n v a l u e E f o r th e

tialenergiesthereare twodeterminate

thatistosay therearetw of inite

ifmbepro ectedf romanypointo fe ither

/ 2 ( E - V ) ] i n t he d i re c ti o n e i th e rw ar d s

e itspathw illbee actly thatcur e .

cases thereareonlytwosuch periodic

ousthattherearemorethantwoinother

mp le

2y 2+ c 2 y2 .

h a e

x = O )

s ( , t -f }

s. WhenEisinf inite ly smallw eha e

any finiteva lueofEw eha eclearlyan

entalmodes ande erymodediffers

fundamentalmode.Toseethislet

y p oi n t N in O X i n a d ir e ct i on p e rp e nd i cu l ar t o O X w i th a v e l oc i ty e u a l to \ / 2 E - a2 O N 2 . A f te r a

ofcrossingsandre-crossingsacrossthe

particlew illcrossthislineverynearlyatright

N . V ary theposit ionofNv eryslightly

andre-pro ectmf romitperpendicularly

y t i l l( byproper" tria landerror method

af tersti l lthesamenumberofcrossings

sese actlyatrightanglesatapo intN ,

. Letmcontinue itsj ourneya longthis

asmanymorecrossingsandre-crossings it

a n d cr o ss O X t h er e e a c tl y a t ri g ht

mNtoN ise actlyhalfanorbit

mainingha lf .

isasmallnumeric theparto f the

ssedby_c 2y2isverysmallincomparison

. Hencethepathisate ery timevery

wo primaryfundamentalnotesformu



s s_ u s e # p d




aninterestingproblemispresented to f ind

v a r ia t io n o f pa r am et e rs ) a e b f s l ow l y

suchthat

y -b s in ( / t- f ,

y = b/ c os ( t - f ,

ton orapractica lappro imationto it.

ossibilitiesinrespectto thiscase

r y sm al l s e em s t ho r ou g hl y t o co n fi r m Ma w el l s

uotedin~ 10 andthatit iscorrect

esmallorlargeseemse ceedinglyprobable

robablethatMa w ell sconclusion w hich

po intmo inginaplane is

.y2................. 1

sf rom 3 2. It iscertainlynotpro ed.

ceptthee uationofenergy

. .. . .. . .. . .. . .. . 2 ,

matica lw or o fpp. 722-725 w hichis

proof forit . Henceanyarbitrarilydraw n

edforthepathw ithoutv io latingthe

toMa w ell sin estigation andw emay

hsuchastosatisfy (1 , andcur esnot

a lltra ersingthew holespacew ithinthe

c 2 y2 = E . .. .. . .. . .. . .. . .. . 3 ) ,

e ll sfundamentalassumption( ~ 10 .

uestionisillustratedbyreducingit

uestionregardingthepath thus:calling0theinclinationto x ofthetangenttothe pathatany

e v e l o ci t y in t h e pa t h w e h a e

= q s i n 0 . .. . .. . .. . .. . .. . . ( 4 ,

q = V { 2 ( E - V ) } . .. . .. . .. . .. . .. . .. . .. . .. 5 .

o ta llengthofcur etra e lled

s 2 q d t = f S 2 ( E - V ) } c os 2 O d s. .. .. . 6

15becomes Isorisnot

} co s2 0 = S ds V { 2 ( E - V ) } si n2 0 . 7 ,



s s_ u s e # p d



C tobea llmo ingtoandf ro . The

dthee ua lbodiesA andC onitstw o

a n d k e e p e u a l t h e a e r ag e k i n e ti c e ne r gy

oeandaf terthesecoll isions tothea eragek ineticenergyofC . K

sofA beinginthespace

cludedinthea erage

of thepotentia land

e ua ltothea erage

utthepotentialenergy F

he s p ac e H i s p o s it i e , A

oursupposit ion thev e locity

e ery timeof its - - € ” H

andincreasedtothe

gmotionf romK toH.

ineticenergyofA isless

ticenergyofC

ectlyrepresentati ek indforthetheoryof temperature

ofthe assumption

so lidorli u idis

ineticenergyperatom

outasaconse uence

andw hich be lie ed

hasbeenlargelytaught

safundamentalpropositioninthermodynamics.

ppro imately

tistosay anassemblageof

oleculemo esfor

esinlinesv eryappro i- L

perienceschangesof

ncomparati e lyveryshortt imesof

y forthek ineticenergyof thetranslatory

gas " thatthetemperature ise ua ltothe

ypermolecule asf irstassumedbyWaterston

e andfirstpro edbyMa w ell.



s s_ u s e # p d



yisdiminishedwhenthesystempasses

ered fromaconfigurationortheconfigurationsuch thatpassagetoanyotherpermittedconfiguration

of thek ineticenergy. B y" tota lenergy o f

nwill bemeantthesumof itsk inetic

t ion. F ore erygi env a lue E

reisafully determinateorbitsuchthat

tionalongit atanyconfiguration

otalenergy E itw il lcirculateperiodica lly

m supposethenumberof f reedoms

n Q , is fully specifiedby igi env a lues

ti ely.Supposenowthesystemtopass

on Q attw otimesseparatedbyan

ethesamev elocit iesanddirectionsof

epaththustra e lledinthisinter a l: isanorbit andit isperiodically tra e lledo erinsuccessi e

oT . Tof indhow toprocurefulfi lmentof

systembestartedfromanyconfiguration

sforthe iv e locity -components( orratesof

f the icoordinates . Tocause itto

u n n o wn t i me T w e h a e i - 1 e u a ti o ns

sei-1of itsv elocity-componentsto

satthesecondasatthef irstpassagethrough

a ti o ns t o s at i sf y a n d i n v i r t u e of t h e e u a ti o n

ingv e locity -componenta lsomustha ethe

times. Thatthetota lenergymayha e

E w eha eanothere uation. Thusw e

tions amongcoordinatesandv elocitycomponents. Eliminateamongthesethe iv e locity -components

uationsamongtheicoordinateswhich

ryandsufficientto securethatQ isa

f tota lenergyE. B e ingi-1e uations

hey lea eonlyonef reedom thatistosay

path o fw hich inthe languageof

eometry theyarethee uations. Theor

n orbitoftotalenergy E.Thusis

f~ 4.



s s_ u s e # p d



A F I N IT E S YS T EM

rminateproblemoffindingan

hastheprescribedva lueE is ingenera l

differentperiodsfortheinfinitenumber

nedbyit.

onlytwofreedoms willhelp

f~ 6fore erycase o fanynumber

ointeddoublependulumconsistingof

B : o n e ( A s u pp o rt e d on a f i e d h or i o n ta l a i s I t h e ot h er ( B ) s u pp o rt e d on a p a ra l le l a i s J , f i e d

implicity letG thecentreofgra ityof

hetw oa es. C a llHthecentreofgra ity

betweentheplaneI andthevertica l

w esha llcall IV ; andletrbetheangle

ndthevertica l. Thecoordinatesand

minanycondit ionofmotionarep q , cb + .

system ink ineticunits willbegW z ,

mof themasses andz theheightof the ir

yconf igurationof thesystem abo eits

be placedinanyparticularposition

iredto findw hatmustbetheposit ion

wh a t v e l oc i ti e s 4 0 4 0 w em u st s t ar t A a nd

thefirstt ime9phasaga inthesamevalue d0

madeone completeturnineither

hallbew holly inthesameposit ion( q 0 k 0

am e v e l o ci t y ( 0 o j 0 ( i n t he s a me d i re c ti o n

hebeginning. Thisimpliesonly tw oe uations

= r 0 o r & l t = c o ( b e c a us e e it h er o f t he s e

ueof thee uationofenergy . A ndw e

a bl e s J 0 a n d ei t he r f 0 or 0 o ( t h e gi e n t ot a l

er40or0ow hentheotherisk now n .

nateproblemis clearlypossible unless

tgenera llyuni ue. Wemayha e

cit iesofA andB startedeachinthe

eachnegati e oronenegati eandtheother

e lo fverygreatmomentof inertia and

mallpendulumhungonacran -pinattached

yw esupposethecran tobecounterpoised sothatthecentreofgra ityofA isinitsa is it isclear

reaterorlessva luegi enforE B may

ytimes beforeAcomesagaintoits



s s_ u s e # p d



utit isclearthat thoughnotgenera lly

o f f indingperiodicmotionwithj ustone

eriodhasno realsolutionunlessEis

nyso lutionsforlargeenoughva luesofE

mberofsolutionsforanyfinite v alue

tionbethat notthef irstt ime but

sthroughitsinit ia lposit ion bothcoordinatesandbothv e locit iesha ethe irprim iti ev a lues. When

etota lenergy isnottoogreat theperiodic

ew illbepurelyv ibratory andthe

utifEbegreatenough A maystill

eB maygoroundandround f irst inone

ther w ithintheperiodofA sv ibration.

tatthefirst andnotatthe second

A acrossitsinit ia lposit ion bothcoordinatesandbothve locit iesha ethe irprim iti ev a lues w emay

alenergy ha estil lw ilderacrobatic

iesgoingroundand roundsometimesin

esintheother. Stillwithanyfinite

afinite numberofmodesforthemotion

thatthethirdtransito fA throughits

hefirstperiod.W ilderandwilder

hin o f if thef irstperiodiscompletedat

andsoon.

nsteinofaproblemisa ll in o l ed

maticalstatementnotincludingany

irst orthesecond orthethird or

ofAthat completesthefirstperiod.Itwill

toarrangesoas tofindatranscendental

eaninf initenumberof f initegroupsof

sof themodesoftheperiodic motions.

typresentsavastly simplerproblem

has nodoubtbeenmanytimesfound

nsintheCambridgeSenate-houseand

minations. Thecharactero f theso lutionof

mic problems isindependentof theabso lute

ergy andof0o. Itdependsonlyonthe

w hichofcoursemaybeeitherposit i e



s s_ u s e # p d




ra lso lutionf -k isclearlyaperiodic

dourquestionofperiodicity re lati e ly

Ireso l esitse lf intothis: -Duringthe

of f -b isthechangeof~ e itherz eroor

ewith27r Acorrespondingq uestion

nwhichour" system isf ree inspace

es andw ithnodisturbingforcef romother

mple inthequestionof rigorousperiodicity

diessuchastheearth moon andsun

mutuallyattractingbodies suchasthe

sideredpresently.

rdinarycloc w ithw eight andpendulum

nt affordsaninterestingillustration.F or

perfectly f le ibleandine tensible le t

edontheshaf to f theescapementwheel le ttheescapementberigidly f i edtothependulum

rigidbodyonperfect k nife-edge

irtually tw obodies eachw ithone

nt-wheel cord andw eight B the

m.Eachimpactoftoothonescapement

ndw atch fo llow edbyamutua lreco il. This

practicalcasesgoessofar asto

tion followedbyse eralmoreimpacts

ohescapes andthecorrespondingne t

eoftheescapement.B utthereisa

sandslipping bothonthenon-wor ing

esof theescapement. The lossonthe

dispensedw ith: butthe lossonthenonwor ingfacesisessentia lto thegoingof thecloc . Inour

upposeeachreco iltoe actly re ersethe

ndescapementin thedirectionperpendiculartothecommontangentplaneof thetwosurfacesattheir

supposethesurfacesto beperfectly

nfinitelygreat mutualforceattheinstant

ly inthatdirection. The j umpingaction

epstoppingthecloc andlettingitgoon

e entanyregularityofgoing.ThereforeIadd thefollowingarrangementofenergy-recei erstoannul

w or ingfacesof theescapement: -P ro long

ent-wheel andf i onit inhe lica lorder



s s_ u s e # p d



DYNA MIC S [ 52

rry ingatitsendadis , w ithitsf ront

ea is. A d usttheescapement-wheel

heneachof itsthirty teethstri esoneor

f theescapement thelowestoneof

tsf rontfacevertica l. Onahori onta l

onoftheshaftplace si tylittleballs

insuchposit ionsthateachshallbestruc bya

rrespondingtoothtouchesthecorresponding

t. Letthemassofeachba llbee ua lto

a l en t o f A ( t h e es c ap e me n t- wh e el & a m p c .

Eachba llstruc by itrecei esthew hole

hadbeforethe impact andlea esA

theescapement-wheelpressingonanonw or ingfaceof theescapement. F i si ty rigidstopstopre ent

i rc u ms t an c es ( ~ ~ 1 - 1 6 , g o in g t oo f a r

chthey areinitiallyplaced.Each

otted toa llow theproperdis o f the

etheballand afterwardspassclear

rm. F orbre ity thesefor edstops

ops. F i a lsosi tyotherstops( f ie ldstopswesha llca llthem insuchposit ionsthattheballssha ll

eouslyandate actly the instant( ~ 12

sthebottomof thecloc -case.

dulumofouridea lcloc w ithits

nearly tothetop tobestartedw ith

eepgo ing. F orsimplicity letthis

ecure thatwhentheweightisrun

angeofv ibrationwillst i l lbewithinthe

tionof theescapementmechanism.

-casebearigidhori ontalplanef i ed

wor bearingthew heelandpendulumin

thatw henthew eight inrunningdow n

umisate itherendof itsrange. The

the impact andthecloc goesbac w ards

turnhomef romtheirf ie ld-stopsate actly

thecord-drum: TV thedri ing-weight: k theradiusof

ghtof thewholerotatingbodyconsistingofcord-drum

af t and60armsanddis s: athe lengthofeacharm

stothepo into f itsdis w hichstri estheba ll. The

a l en t i s ( T r 2 + w 2 / a 2.



s s_ u s e # p d




nge actlye erysteptil lthew eight

yof thependulumandofthereturning

assesthroughitsinit ialposit ion. If it is

stri esaga instanunad ustedstop

edforatime w iththependulumv ibrating

llrange andonetoothofthewheel

r ingfacetofthe escapement:but

erysoon thetoothw illescape thecloc

hew eightw illrundow n andagainstri e

romit thist imenotw henthependulum

herendof itsrange.

iteorderlyactionw illf o llow and

oothwillbehoo edupby theescapement

ac w ardsabeatortw o butaf terav ery

one itw il lgo forw ardtil lthew eight

n. " Soonerorla ter thebottomw illbe

hependulumisv erynearlyatrestat

dw hense era lenergy - rece i ersare in

ehomeandstri edis satrighttimes

ac w ardsforagoodmanybeats.

thatistosayaf tersomef initenumber

ars thew eightw illstri e thebottom

erynearlyatrestat eitherendofits

ba llssoverynearly stri ingeachits

w illbedri enbac , w indingupthe

sthetopstop andimmediate ly ora f ter

ginsaga intogoforw ardandlettheweight

ectisnotthefortuitousconcourseof

otionofafinite system.

otheendof~ 12 letthetopstop

sstruc by thew eightataninstantw hen

dofitsrange.The cloc instantly

ndgoesonretracinge erystep and

thenumerousimpacts o f itsf irstforw ard

stri esthebottome actlywheneachof

ngitsf ield-stop andw henthependulum

thesameendof itsrangeasw henthe

mthefirsttime.Thusaperfectlyperiodic

.



s s_ u s e # p d



-periodsinw hichthecloc is

eweightrunningdow n anymoderate

slightblow onthependulum oraho lding

stoppedforsometime largeorsmall

edifference inthesubse uentmotion: t il l

tomofits range whenwefindthat

ndthestateof thingsdescribedin~~ 1 ,

nysuchdisturbanceduringaha lf -period

gbac w ardscausesthebac w ardmotion

rdmotionto followimmediately or

aterorless numberaccordingasthe

glyinfinitesimalorbutmoderatelysmall.

ustrationofthe" dissipationofenergy "

nityofattemptswhichha ebeenmade

inciple " or" theSecondLaw ofThermodynamics " ortheoriesofchemicalactiononLagrange sgenera li ed

lemof thethreebodies intw o

eLunarTheory " secondly " theP lanetary

theSun isineach casev astlylarger

ers.Inthefirstcase thetwoothers

aresonearoneanotherincomparison

fromeitherthat hisforceproducesbuta

ati emotionoftheEarth andMoon

raction.Inthe secondcase two

yunderthe Sun sinfluencewithcomparati elysmalldisturbancebytheirown mutualattraction.In

simplicity neglectthemotionof theSun s

considerhimasanabso lutely f i ed' centre

artheory supposethecentreof

h an d M oo n t o mo e v e r y ap p ro i m at e ly i n

w( withoutnecessarilyconsidering

a llerthantheEarth ataninstant

throughS gi ee ua landopposite

thelineMEso astoannultheir

ne if theyhadany andtocauseeachto

dicularly to it. If thene ttimetheirl ine

gain mo ingperpendicularlytoME



s s_ u s e # p d




oSIis rigorouslyperiodic.Thiswesee

motionsarere ersedatanyinstant

racethe irpaths andif suchare ersa l

rpendicularlycrossingthelineSI the

o thedirectpathswhichare traced

a l.

iesbegi eninline SME w e

oftheirmotionif wepro ectthem

endiculartothis linewithe actlysuch

ttimeMEisagaininlinew ithS now

motionareagain perpendiculartoEM.

as threesolutions inoneofwhich

ectionaresogreatthatMandEarecarried

inoppositedirectionsroundtheSun

oneanotherandinline onthefarside

iscaseweha ecerta inlyonly tw o

scribese ceedinglynearlyacircle

andEmo ere lati e ly tothepointI

hatappro imately incircles buttoa

ntheellipsescorrespondingtothe lunar

aria tion andquiterigorously intw o

r eseachdifferingv ery little f romthe

entreofthev ariationalellipseisatI:

pendiculartoSIande ceedstheminora is

9 6 b e in g t he s u a re o f t he r a ti o ( 1 / 1 ' 4

ofSIto theangularv e locityofME each

y fi eddirection. Therearetw oso lutions

fw hich( asintheactua lcaseofEarth

mewardsas intheothercontrary -wards

etimesasgreat asitis when

e SME andotherdimensionsthe

a easo lutionforperiodicitycorresponding

iththeorbitalcur esofMandEroundI

mcirclesand largelyfromellipses.

erta inlimit thisk indofso lutionbecomes

whollyuninterestingto followthe

esroundIfor increasingmagnitudes

hesolutionreferredtoandre ected



s s_ u s e # p d



beandbecomesnow more interesting but

orrespondingsolutioninwhichMand

a r e pr o e c te d s o as t o r e o l e i n t he s a me

otionof tw oplanets. Gi enSV E

oplanetsatdistancessuchasthoseof

isre uiredtopro ectthemwithsuch

uentmotionisrigorouslyperiodic.A

pro ectingthemperpendicularlyto

esthatthe irperiodsof re o lutionroundS

u a l a n d e a c tl y s uc h t ha t a t th e n e t t i me

ew ithS themotionsarerigorously

Thevelocit iesw hichmustbegi en

tbesuchthatthema ora esof the

escribedareappro imatelye ua l. This

be longsrathertotheC ometary thantothe

perpendicularly toSV E w ithsuch

egi ennumberof t imesof the irbeing

e irmotionsare forthef irstt imeagain

hedeterminatev elocitieswhichfulfil

besuchthatthe orbitsareappro imatelyellipsesofeccentricitiesnotdifferingmuchfrom those

e ma o r a e s s uc h t ha t t he p e ri o ds h a e t h e

torenderthelineofthe threebodies

arcrossingappro imatelycoincident

perpendicularcrossing.

P E RI O D I C M O T I O N B E I N G A C O N T IN U A T I O N

I O D I C MO T I O N O F A F I N IT E CO N S E R V A T IV E S Y ST E M.

. No . 26 1891 Phil. Nag. Dec. 1891.

' , . .. b e g en e ra l i e d c oo r di n at e s of a s y st e m

, f , . .. b e th e ac ti on i n ap at h ( ~ 2 a bo e

, ' , . . . t o t h e c on f ig u ra t io n ( I s . . .

E-V ) w ithanygi enconstantva lueforE

e in g t he p o te n ti a l en e rg y ( ~ 3 a b o e , o f

enfore erypossibleconf igurationof the



s s_ u s e # p d



RI O D I C M O T I O N

. . an d v ' , 4 , v ' , . . .. b e th e ge ne ra li e d

ofthesystemasitpassesthroughthe

b . .. a nd ( ' ' , I , . .. r es pe ct i e ly . I f by a ny

o edtheproblemof themotionof thesystem

e * ( o f w hi c h V i s t he p o te n ti a l en e rg y , w e

e n s et o f v a lu es o f r p . .. f , s b , . .. t ha t is

f un ct io n of ( 4 c q , . .. ' , b , . .. . Th en b y

ThomsonandTait sNaturalPhilosophy

w eha e

dA

= ' dX ' dr . 1

dA

- dd~ d ' = -d d ' "

ateaparticularpatht-from position

' , . .. w hi ch f or b re i ty w es ha ll c al l P , t o po si ti on

w h ic h w e sh a ll c a ll P . L e t o P o P b e a pa r t of a

f rom w hichP P isinf inite ly litt ledistant.

oP isperiodicornot pro idedit isinfinitely

i d ed O P a n d oP a r e in f in i te l y ne a r to P

w e ha e b y Ta yl or s t he or em a nd b y( 1 ,

, oX , * * . oo , X ' , . ..

+ ( ~ - + ... -o ( ' - o ' ) - o ' ( St - o ) -...

o ) ( , - + } + .. .. .. .. .. 2 .

edbymybrother P ro f . JamesThomson todenotea

u e of E t h e to t al e n er g y ( ~ 3 a b o e , t h e pr o bl e m of f i nd i ng

P toanyposit ionP isdeterminate. Itsso lutionis for

em adeterminatefunctionofthecoordinateswhich

t thetimerec onedf romtheinstantofpassingthroughP .

hecaseofa particlemo ingundertheinfluenceof no

nganinfinitestra ightline . F orasingleparticlemo ing

niformforce inpara lle ll ines( asgra ity insmall- sca le

eso lutionisduple orimaginary. F ore eryconstra inedly

isinf inite lymultiple asisv irtua llyw ellk now nbye ery

ofaB osco ichianatomflyingaboutwithinanenclosing

ery tennisplayerfortheparabo lasw ithw hichheisconcerned

mw allsorpa ement.



s s_ u s e # p d



YNA MIC S [ 52

choosingourcoordinatessothat

& a m p c . a r e ea c h z e r o f or e e r y po s it i on o f t he

foranyposit ionof thispath betheaction

eroatoP . Theseassumptions e pressed

ws: dA _ dA dA dA dA o

o = 0 - 0 l

#- d = ' d X / '

, i fi = 0 X = 0 . .. i = 0 X ' = 0 . .. . .. .. .. .. 3 ) .

o = , 0oX = 0 . .. O = , o 0 = , o X ' = O . .. ( 4 ;

o , o , o X ' , .. ) = A ( 0 0 , . ..0 , 0 ... ... 5

an d of ( 3 ) a nd ( 1 , ( 2 b ec om es

, ' , o ... o 00

+ 66+ + " '

1 4 0 0+ 1 5 O t + 1 60 '

+ 2 5X X + 26% X ' . . 6 ,

+ 3 6 5

/

mplicityo fnotation w esupposethetota l

system thatistosay thetotal

es * , & l t , X , i - t o b e fo u r a n d f o r

b y ac c id e nt i n ( 6 a n d ( 8 a s s ub s cr i pt s

2A _

= 1 2 o = 22 & a mp c .. .

d b y( 1 ,

( ' + 1 44 + 1 5X % + 1 6̂ '

- + 2 40 + 2 5 X ' + 2 69

3 3 P + 3 4 0 + 3 5 X I + 3 6

. + 4 4 ' + 4 5% + 4 6

4 52 + 5 X + 5 4 + + 5 6

2% + 6 4 + 6 4 x + 6 6 + 6 6



s s_ u s e # p d




stodeterminethethree displacements

e t hr e e co r re s po n di n g mo m en t um s | , q , ' , f o r an y

t er m s of t h e in i ti a l v a l u es b , X ' , C ,

s up po se dk n o wn .

supposition( ~ 24 that0P oP is

letQ beapositiononitbetw eenoP

t o a o i d am b ig u it y c a ll i t o P Q o P .

entoco incide inaposit ionw hichw e

r ds l e t oP Q o P o r O Q O , b e t he c o mp l et e

t as w e h a e c a l l ed i t ( ~ 2 a b o e . O u r

infinitelyneartothisorbit andP and

osit ionsinitforw hichA hastheva lue

sareinfinitelynearto oneanotherand

i andO i+ l consideringthemasthe

hich risz ero forthe ithtimeand

e f romanearlierinit ia lepochthanf irst

hichweha ebeenhithertoconsidering.Itis accordinglycon enientnowtomodifyournotationas

= x i ' = i r ' = = , = ' ) l

l = ri + l = + i 7 7= + i = ? i l. .. .. .. .. 9 .

ethegenera li edcomponentsofdistance

tthroughr= 0 o f thesystempursuing

heorbit and:i q i ' ia rethecorrespondingmomentum-components. Withthenotationof (9 ,

mee uationsbyw hichthev a luesof these

thtimeof transitthroughF = 0canbe

forthe ithtime. Theyaree uations

daretobetreatedsecundu martem as

= ' P X i i + = pi . ( 1 0 .

P7 w ? i + l= P "

ncontinua lly increasesa longthepath. Whatismeantis

stant asregardsaction byanamountw hichise ua lto

tionintheperiodicorbit. Thesub ectmaybeelucidated

orays oflight wherethereareonlytwocoordinates

iioninaplanetrans ersetotheray : cf . ThomsonandTa it s



s s_ u s e # p d



( 8 modif iedby (9 , andeliminating

+ ( 1 2+ 1 5 + 4 2 p+ 4 5 X

46 = O

( + ( 2 2+ + 5 5 2p + 5 5 ) .

56 = = 0

64 + ( 3 2 + - + 6 2p + 65

66 s= 0

4 12= 21 & amp c. w eseethatthedeterminantforthee lim inationof theratiosIX 19issymmetrical

Henceitis

p 2p -2 C ( p + p -l + 2 Co.. . 1 2 ,

C arecoeff icientsofw hichthev a luesinterms

c. areeasilyw rittenout. Thisdeterminante uated

uationof the6thdegreefordeterminingp

re isanothere ua lto itsreciproca l.

a tionof thethirddegreebyputting.

.. . .. . . .. . . .. . 1 ) .

e t he r o ot s o f th e e u a ti o n th u s fo u nd . T he

pare

~ ( e -1 ; es + V ( e 2- 1 . .. 14 .

any rea lv a luebetw een1and-1 it is

cosa+ tsina . . .. . . .. . . .. . . .. . ( 15 .

- sina



s s_ u s e # p d




rthef irstt imeofpassingthrough

atesandthreecorrespondingmomenta

r 7n 1 t ob ea ll gi e n w e fi nd

p l- i + + A 2 p 2 + A p -+ A p i + A p -i

' p&gt - i+ B 2p2i+ B 2 2- i+ B 3 p i+ B 3 ' pi. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . ( I6) , . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .

' p 1- i+ F 2 p 2i + F 2 p2 -i + F 3 p i+ F 3 ' p i

A 2 . . .F 1 F ' , F 2 F 2 a re 3 6 c oe ff ic ie nt s

t he s i e u a ti o ns ( 1 6 w i th i = 0 :

8 , m o d if i ed b y ( 9 ; w i t h i su c ce s si e l y

4 5 w it h th e gi e n v a lu es s ub st it ut ed f or q b X I

i n th em a nd w it h fo r 02 X 2 & a m p c . th ei r v a l ue s by ( 1 6 .

sthate erypathinf inite lyneartothe

ery rootof thee uationforehasa

d-1. Itdoesnotpro ethatthemotion

n isfulfilled.Stabilityorinstability

stedwithoutgoingtohigherordersof

siderationofpaths v erynearlycoincident

mb e r 10 1 8 91 .

motionand itsstabilityhasbeen

byM. Po incare inapaper " Surle

tlese uationsde ladynami ue " for

a esty theK ingofSw edenw asawarded

1889. Thispaper w hichhasbeen

sA ctaMathematica 1 , 1and2

oc h o lm 1 8 90 o n ly b e ca me k n o wn t o m e tw e l e

yley.Iamgreatlyinterestedto find

thesub ectofmycommunicationof

Society " O nsomeTestC asesforthe

nDoctrineregardingDistributionofEnergy ;

thefo llow ingparagraph:- O npeut

o isinaged unetra ecto ire fermeerepresentantunesolutionperiodi ue so itstable so it instable i lpasse

ecto iresfermees. C e lanesuf fitpas en

nclurequetouteregiondelespace sipetite



s s_ u s e # p d



a erseparuneinf initedestra ecto iresfermees

ra cettehypotheseunhautcaractere

Thisstatementise ceedingly interesting

w ell sfundamentalsupposit ionquotedin

thatthesystemif le f tto itself in itsactua l

soonerorla ter passthroughe eryphasew hich

uationofenergy t" anassumptionw hich

aconclusion butasaproposit ionw hich" w e

fidenceassert ...e ceptforparticular

ef i edobstacle . Itw il lbeseenthat

sis ha ingahighcharactero fprobabil ity "

w ell s w hichassertsthate eryportion

nalldirectionsbye ery tra ectory . The

in~ 1 + , asseemingtomequitecerta in

ersinfinitelylittlefrombeinga fundamental

ecessaryconse uenceofMa w ell sfundamental

whichstill seemstomehighlyprobable

casesareproperlydealtw ith.

tatement pp. 100 101: - IIyaura

titesa2distinctes.Nouslesappellerons

dela solutionperiodi ueconsideree.

nttousree lsetn6gatifs laso lution

carlesquantites4ietqiresterontinf4rieures

sentendrecemotdestabiliteau sens

a onsneglige lescarresdes: etdesq, et

entenantcomptedecescarresleresultatne

ouspou onsdireaumoinsquelesa

airementtrespetits resteronttrespetits

Nouspou onse primercefa iten

eriodi ue j ouit sinondelastabilite

astabil ite temporaire . Heretheconclusionof~ 3 1ofmypresentpaperisperfectlyanticipatedand

nterestingmanner. M. Poincare sin estigationandmineareasdif ferentastw oin estigationsof the

llbe andit isverysatisfactory to f ind

usions.

ermee o fM. Po incare isw hatIca lleda" fundamenta l

icmotion or" anorbit.

o l . ii . p . 71 4 . + [ S u pr a p . 4 92 .



s s_ u s e # p d



NPLA NEKINETIC TR IGONO METRY

S S S T H EO R E M O F C U R V A T U R A I N T EG R A.

M a ga i n e V o l . x x x i i . No . 1 8 91 p p . 47 1 -4 7 .

beautifultheoremof the" Spherica l

rigonometry publishedabout16 7 and

slaterbyGeneralR oy inthetrigonometricalsur eyof theB rit ishIsles w assplendidlye tendedby

f the" C ur atura Integra. Theremust

minthe" k inetictrigonometry suggested

nandTait sNaturalPhilosophy

b ( c ( d , f o r th e mo ti on o ft he g en er al i e d co ns er a ti e

umberofvariables. F orthev erysimple

mo inginaplane it iseasilyw or ed

inendea ouringtow riteacontinuationof

a lMaga ine O ctober onthePeriodic

Genera lescircaSuperf iciesC ur as auctoreC aroloF rederico

e[Gottingensi oblatseD. V III. O ctobr. MDC C CX X V II. C o llected

G 6 tt i ng e n 1 8 7 . T h o m so n a nd T a it s N a tu r al P h il o so p hy



s s_ u s e # p d



m w hichIhopemaybeready toappear

Hereis thetheoremmeantime.

, R C A Sbethreepathsofaparticle

e underinf luenceofaforce( -d - )

threeplacesin anydirectionintheplane

athesumof thek ineticandpotentia l

a lue( E ineachcase. Thesumof the

C e ceedstw orightanglesbyanamount

adians ise ua ltothesurface- integra lo f

t h ro u gh o ut t h e en c lo s ed a r ea A B C V 2 d e no t in g

dy2.

ma r t h a t

cionof ( x , y , f fd dysurface- integration

dsline-integrationallrounditsboundary

ria tionof * inthedirectionperpendicular

t. Hencethesurface-integralmentionedin~ 2ise ualto

. . .. . . .. . . .. . . .. . . .. . . .. . . 1 .

( 1

rmal-componentforce( N w esha llca ll it ;

e s u a re o f t he v e l oc i ty ( v 2 w e s h a ll c a ll i t .

.. . . 2 .

a t ur e ( 1 / p w e sh a ll c a ll i t , a t a ny p oi n t in

A B , B C C A . He n ce d i i d in g f ds

ngrespecti elytothesethreearcs

d d s d s , w e fi n d fo r ( 2 ,

directioninthearcA B , andsim ilarly

e theorem.



s s_ u s e # p d



Y O F P E RI O D I C M O T I O N .

c i at i on R ep o rt 1 8 92 p . 6 8 ( t i tl e o nl y ; N a tu r e

1 89 2 p . 3 8 4.

stigationofthissub ectwasillustrated

hichasimpleharmonicv erticalmotionwas

pporto fapendulum. Whentheperiodof

sonehalfofthat ofthenaturalmotion

uil ibriumbecameunstable andthe

edthev erticalmotionofthebobto be

motionofincreasingamplitude.Ifthe

owlessened thev erticalmotionagain

rodpoisedv erticallyinunstable

mestablebyha ingitspo into f support

monicmotion o fproperperiod inav ertical

remar edthatitw asw ellk now nto

re o l ingshaft w hendri enatacertain

andmighte enbrea , thoughathigher

comestraight. LordK el inhadnow

" O ntheMaintenanceofV ibrationsbyF orcesofDouble

P h il . M ag . V o l . x x I . 1 8 87 p p . 14 5 -1 5 9 S c i e nt i fi c P ap e rs V o l . Im .



s s_ u s e # p d



[ 5 5

T IONO F DYNA MIC A LPRO B LEMS.

c i at i on R ep o rt 1 8 92 p p . 64 8 -6 5 2 N a tu r e V o l . X L V I .

8 5 3 8 6 P hi l. M fa g. V o l . x x x i . p p. 4 4 â € ” 4 48 .

eridianalcur esofcapillarysurfaces

edinPopularLecturesandA ddresses V o l. I.

2 andil lustratedbyw oodcutsmadef rom

or edoutaccordingto itw ithgreatcare

errywhena studentintheNatural

ow Uni ersity suggestsacorresponding

ynamicalproblems.

ardingthemotionofa singleparticle

efo llow ingplanfordraw inganypossible

a forceofwhichthepotentialis gi en

ane. Suppose fore ample it isre uired

lepro ected w ithanygi env e locity in

ughanygi enpo intPO (f ig. 1 . C a lculate

ceatthispo int anddi idethes uare

a lue to f indtheradius

hatthatpo int. Ta ing P2

sses f indthecentreof Q 1p0

he l i ne P o , p e rp e nd i cu l ar t o

ughPo anddescribea

m a i ng P Q , e u al t o ab ou t

rthe secondarc. oC

locityfortheposition

potentia llaw and asbefore K

eshradiusofcur aturefor

componentforceforthe L

andfor thev elocity

tionofQ , . Withthis F ig. 1.

o f thecentreofcur ature C , inP1C oL

ughP1. Withthiscentreofcur ature



s s_ u s e # p d



T I O N O F D Y NA M IC A L PR O B L E MS

r ature describeanarcP1P2Q2ma ing

a lf the lengthintendedforthethirdarc

atureforpositionQ2 draw anarcP2PQ3 ;

e. Thisprocessiswelladaptedfor

ia landerror methoddescribedinmy

TestC ases o f theMa w ell-B o lt mannDoctrine

Energy " ~ 1 ; P roc. Roy . Soc. J une11

.

e( fig.2 hasbeendrawnwithgreat

nterestingsuccess inthe" tria landerror

nd simplestorbit bymysecretary

orthecaseofmotiondef inedby the

enononeof thelinescuttingthe

~ , andatf irstatarandomdistancef romthe

sw or edaccordingtothemethod



s s_ u s e # p d



dw asfoundtocutthea iso fx atanobli ue

es w ithunchangedenergy -constant w ere

sat greaterorlessdistancesfromthe

asfoundtocutthea iso fx perpendicularly .

partof theorbit andisshowninfig. 2

erto completetheorbit whichis

idesof thea iso fx andy .

motionrelatedtotheLunarTheory

oonbe infinitelysmallincomparison

h andtheearthandsuntoha euniform

heircentreofgra ity . Let( x , y be

e lati etoO X inlinew iththesun outw ards andOYperpendicularto it inthedirectionof theearth s

now ne uationofmotionre lati e ly to

g i e s f o r th e e u a ti o ns o f t he m oo n s m o ti o n

rom0( theearth o f thecentreofgra ity

,

t d . .. .. .

,

,

of theattractionsofthesunand earth

heangularve locityof theearth sradius ector. F romthiswef ind forthere lati e -energye uation

+ a 2 + y 2 - . .. .. .. .. 3 )

ant andforthere lati e -cur aturee uation

d t N d t2

” ~ -- -- ~ ~ ~ ( 4 ,

( d z + dy2 2 d 2 + dy2

onentperpendiculartothepath ofthe

, w it h

_C dV ( 5

. .. . . .. . . .. . )

( 6 .



s s_ u s e # p d



L U T I O N O F D Y N A MI C AL P R O B L E M S 5 1 9

n sv e locityandptheradiusofcur atureof

to t h e re o l i n g pl a ne X O Y w e h a e

a 2 y - V . .. .. .. .. .. .( 7 ,

. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . 8 .

ss andahisdistancef romtheearth

smassinfinitelysmallin comparisonwith

......... 9 ,

1 V = 2 a + m .. . .. . .. . . 1 0 ,

y2 2 r

h s m as s a n d r = V / x 2 + y 2

2 a 2 - 2 a 2 + + m . .. . .. . .. . 1 1 .

= l a n d m = b , f o r si m pl i ci t y in t h e

f llows w eha e

2+ - . . . . . .. . . .. . . .. . . .. . . .. . . .. . 12 ,

. . .. . .. . .. . .. . .. . . 1

.................................. 15 .

2 a nd ( 1 ) , G . W . Hi ll h as w it h fo ur

foundx andye plicitly intermst forthe

ase whichgi esthesimplestorbit

o l i n g pl a ne X O Y ; o f wh i ch t h e on e wh i ch

iationf romthewell- now n" v aria tiona l

unartheory isasymmetrica lcur ew ith

gcuspscorrespondingtothemoonin

supposedthistobethemoste tremede iationf romthevaria tiona lo a lpossible foranorbitsurrounding

hisMethodesNou ellesde laMdcani ue



s s_ u s e # p d



2 , admiringj ustly themannerinw hichHill

ment" studiedthesub ectof f initeclosedlunarorbits po intsout

respondingto

ding Hil l s w rongly

uspedorbit. MrHillte lls

ticism.Thelabour

ccurateanalytical

e sloopedorbits by

dprobablybev erygreat.

t itm ightinterest

stoapplymygraphic --

t leastoneof }

ts inourPhysica l( and

ry intheU ni ersityof /

presentsa loopedorbit

outaccordinglybyMr7

eO f f icia lA ssistanto f

losophy fromthe

1 5 a b o e . T he i n it i al v a l ue s

r e w erex = 2

- 1 0 a nd t he re fo re F i g . 3

8 .



s s_ u s e # p d



E V E R Y P RO B L E M O F T w o F R E E DO M S I N

NA MIC STO THEDR A WINGOF GEODETIC

C E O F G I V E N S PE C I I C C U R V A T U R E .

c i at i on R ep o rt 1 8 92 p p . 65 2 6 5 ; N a tu r e V o l . X L V I .

8 6 .

aseof two- f reedommotionispro ed

pondingcaseofthemotionof amaterial

edynamics w ithanygi env a luefor

theresultantve locity q , a tanypo int

ow n fu nc ti on o f( x , y , b ei ng g i e n by t he e u at io n

tentia la t(x , y ; ande eryproblemdepends

f ds( theMaupertuis" action ) isa

t S o f the infiniteplane f indasurface

nitesimaltriangleA B ' C drawnonithasits

acorrespondingtriangleA B C inthef ie ld X ,

odenotingtheva lueofq atanyparticular

theplane. B y theprincipleof leastactionw esee

nS , correspondingtopathsonS are

mniccaseofmotion o faparticleonS ,

ompleterepresentati eofthemotionon

nderforcew ithanyarbitrarilygi enfunction

danyparticulargi env a lue E forthetota l

article .



s s_ u s e # p d



atthesurfaceS , tobefoundaccording

a n d th a t it s s pe c if i c cu r a t ur e ( G a us s s n a me f o r th e

lcur atures atanypo intise ua lto*

n o f th e f in d in g o f S . A s o ne e a m pl e

efulnessofthis methodindynamics

cmotionof anunresistedpro ectileis

eodeticli nesonacertainfigure of

e plicite uationise pressedintermsof

redby thetransformationfromorbitsontheplaneto

S . F orak inetictriangleontheplanethee cessof the

nglesistheareamultipliedbyV 2logq ( cf . supra p. 514 .

stheareamultipliedby theGaussiancur ature. B ye uating

lows. On thesek inetictransformations cf . Larmor P roc.

1 8 84 D a rb o u , T h eo r ie d e s Su r fa c es V o l . II . l i r e v .



s s_ u s e # p d



F " M E RC A TO R S " P R O J E C TI O N P E R O R M E D

CA LINSTRU MENTS.

o l . X L V I . S e p. 2 2 1 8 92 p p . 49 0 4 9 1.

erali ingMercator sPro ectionis

ommunicationtoSectionAofthe

recentmeetinginEdinburgh entitled

ProblemofTwoF reedomsinC onser ati e

fGeodeticLines onaSurfaceofgi en

A nabstracto f thispaperappearedin

mer commonlyk now nas" Mercator

me , ga etothew orldhischart now of

g at i on . I n it e e r y is l an d e e r y ba y e e r y

n e i f n ot e t e nd i ng o e r m or e t ha n t wo o r

orfarthernorthandsouththan a

rthreedegreesof longitude isshow nvery

eshape: rigorously so if ite tendso er

n infinitesimaldifferenceoflongitude.

ointersectinglineson thesurfaceof

orouslywithoutchangein thecorrespondingangleonthechart.

imaginedas beingmadebycoating

bewithathinine tensiblesheetof

fore ample( forsimplicity how e er

tensiblebutinelastic -cuttingaw ay

ttedfromthechart cuttingthesheet

thatof 180~ longitudefromGreenwich

ngthesheete erywheree cepta longthe

ea llthecirclesof la titudee ua linlengthto

e uator andstretchingthesheetinthe

thesameratioasthe ratioinwhich

stretched w hilek eepingatrightangles



s s_ u s e # p d



themeridiansandtheparallels.The

e laidoutf la torrolledup asapaper

edMercator schartforabodyofany

erical isaflat sheetshowingforany

be drawnonapartof thesurfaceof

glineswhichintersectat thesameangles.

te dimensionscanonlyrepresentapart

a finitebody ifthebodybesimply

ay if ithasnoho leortunnelthroughit.

chor ringcanob iouslybemercatori ed

een forthecaseof theglobe thattwo

i ethew holesurface anditw il lbe

ochartssuffice foranysimplycontinuous

ere tremely itmayde iatef romthespherica l

a l fo r 1 84 7 i t s e d it o r L i o u i l le g a e a n

n accordingtow hich if thee uationofany

en aseto f l inesdraw nonitcanbefound

hesurfacecanbedi idedinto inf initesimals uares by these linesandthesetof l inesonthesurface

ngles. Now itisclearthatifw eha e

surfacethusdi idedinto inf initesimal

atistosay di idedinto inf initesimals uares

uarestogether a llthroughit w ecan

toonesi eandlay themdow nonaf la t

t withitsfouroriginalneighbours and

ofsurface ismercatori ed. E ceptfor

o lution orane ll ipso id orv irtually

i ou i l le s d i ff e re n ti a l e u a ti o ns a r e of a v e r y

eonly recentlynoticedthatw ecanso l e

withanyaccuracydesiredifthe problem

w hichit isnot bya idofavo ltmeter

orothermeansofproducinge lectriccurrents

be mercatori edinthin sheet

essthroughout. B y thinImeanthatthe

llf ractionof thesmallestradiusofcur atureofanyparto f thesurface.



s s_ u s e # p d



T IO N O F M E RC A TO R S P R O J E C TI O N

of thesurface N S andapply the

atthesepoints.

ee lectrodesof thevo ltmeter tracean

ascloseasmaybearoundoneelectrode and

ne F , asnearasmaybearoundtheother

setwoe uipotentials E I tracea large

differente uipotentials. Di ideoneof the

F ] i n to n e u a l pa r ts a n d th r ou g h th e

nescuttingthew holeseriesofe uipotentialsatrightangles. Thesetrans erse linesandthe

thew holesurfacebetw eenEandF into

Ma w ell E lectricityandMagnetism ~ 651 .

toonesi eandplacethemtogether

Thusweha eaMercatorcharto f thew hole

ationcorrespondtothenorthand

scharto f theworld andourgenera li ed

fillingtheessentialprincipleofsimilarity

ybeconstructedfor asphericalsurfaceby

opointsnot necessarilythepolesatthe

er. If thepo intsN Sare inf inite lynear

ngMercatorchartforthe caseofaspherical

ographicpro ectionof thesurfaceonthetangent

f thediameterthroughthepoint C

nthis casethee uipotentialsand

sonthe sphericalsurfacecuttingNSat

ingit respecti e ly .

o thersurfacew emaymercatori eany

A B C D b o un d ed b y f ou r c ur e s A B , B C

anotheratrightanglesasfo llows. C utthis

metallicsheet totwoofits opposite

f o r in s ta n ce f i i n fi n it e ly c o nd u ct i e b o rd e rs .

vo lta icbattery totheseborders and

uipotentiall inesbetw eenA B andDC .

w eenconsecuti ee uipotentialsintos uares ,

achdistantf romthene tby thesame

raw cur escuttingperpendicularly thew hole

Thesecur esandthee uipotentia ls



s s_ u s e # p d



YNA MIC S [ 57

nto inf initesimals uares. E ua li e the

getheronthef la tasabo e.

ticalinstrumentsbywhichwecan

satrightanglesto asystemalreadydrawn

hematicalinstrumentsaltogether and

i idingintos uaresbye lectrica linstrumentsasfo llow s. R emo etheconductingbordersf romA B , DC

eborderstoA DandB C applyelectrodes

rs andasbeforedraw ne uidifferent

ondsetofe uipotentials andthef irst

area intos uares.



s s_ u s e # p d



TOR C HA R TONO NESHEETR EPRESENTING

YCO MPLEXLY CO NTINUO U SCLO SEDSU R ACE .

l . X L V I . O c t . 6 1 89 2 p p. 5 41 5 42 .

dbyanyperforation itssurface isca lled

w e ercomplicateditsshapemaybe. Ifa

forations ortunnelst itsw holebounding

pe lycontinuous" ; duple lyw henthere is

n+ 1 -ple lyw hentherearenperforations.

p ofnanchor-rings( or" toroids )

re lati eposit ions isacon enientand

ofan ( n+ l -ple lycontinuousclosed

X X

aticsof thegenera lproblem seeR iemann Gesanmmelte

" T h eo r ie d e r Ab e l s c he n F u n c t io n en ( 2 . L eh r si t e a u s

" Nachlass " F ragmentausderA na lysisSitus a lsoB etti

. i . ( 1 8 70 - 1 a l so F o r sy t h s T h eo r y of F u n c t i on s a n d

adeepho llow notthroughw ithtw oopenends. The

propriatefortheapertureof ananchorring.Neither

be ingune ceptiona llya a ilable Iamcompelledtousethe

n.



s s_ u s e # p d



aq uadruple lycontinuousclosed

hinsheetmeta l uniformastothic nessand

ity throughout. Toprepareforma inga

titopenbetw eenperforationsC andB ,

space inthemannerindicatedat2 1 and

cti eborderstothe twolipsseparated

apply thee lectrodesofav o lta icbattery to

fmo ablee lectrodesofav o ltmetertrace

av ery largenumber( n-1 o fe uidifferente uipotentialclosedcur esbetweenthe+ and-borders.

ee uipotentials intopartseache ua lto

perpendicularlyacrossittothe ne t

ideof it andthroughthedi isiona l

utt ingthee uipotentialsatrightangles.

am-lines. Theyand the( n+ 1 closed

ingthe inf inite lyconducti eborders di ide

minfinitesimals uares ifnmbethe

chw efoundinthee uipotential. The

wthegeneraldirection oftheelectric

f thecomple circuit eacharrow

metalshelloneitherfar ornearside

paper.

tream-linesintheneighbourhoods

e d i n or d er o f t he s t re a m1 2 3 , 4 w e

ipsthereis onestream-linewhich

yononesideandlea esitperpendicularly

cIca llthef lu -shed- line(or forbre ity

the liptowhichitbe longs. Thestream- lines

-shed onitstw osides passinf initely

ofthelip andcomeininfinitelynear to

u -shedonitstw osides. LetF 1 F 2

ownonthediagram bethepo intsonthe+ terminal

s he d s of t h e li p s 1 2 3 , 4 p r oc e ed a n d

bethepo intsatw hichthey fallonthe- lip.

& a m p c . d e no t e th e p oi n ts o n t he f o ur l i ps a t w hi c h

by the irf lu -shedlines.

pre iousarticle ( " Genera li a tionofMercator sPro ection" ) , in~ 3 , andinlastparagraphbutone aremanifestlywrong andmust

htherulegi enfordi idinginto inf initesimals uares in

r ec t ed i n t e t p . 5 25 s u p r a.



s s_ u s e # p d



A RTO F A C YC LICA LLYC ONNEC TEDSU R A C E529

P , 1 , P 4 1 4 p b e th e di ff er en ce s of p ot en ti al

r om S 1 t o T , T t o S 2 . .. S 4 t o T4 a n d T4 t o

nedifferencesofpotential.W eare

Mercatorchart. Wemightindeedha e

orateconsiderationsandmeasurements

eofmypre iousarticle butthechart

inf initecontractionate ightpoints the

T1 . . . S4 T4. Thisfault isa o ided

thewholesurfaceonafinite scalein

thefollowingprocess.

beofthinsheetmetal ofthesame

ityasthatofourorigina lsurface andon

a r f o u r p oi n ts A h h 2 h , h 4 a t c on s ec u ti e

erenceproportionalrespecti elytothe

neswhichwefind betweenF 1andF 2

a nd F 4 F 4 a n d F , o n th e + l i p of o ur o ri gi na l

h 2 h h 4 d ra w l in e s pa r al l el t o t he a i s o f

nte ua ltothetota lcurrentw hich

he -lipthroughthe originalsurfacebe

esentcylinderbya v oltaicbatterywith

sonthe cylinderv eryfardistantonthe

Mar onthecy lindere ightcircles

t d is t an c es c o ns e cu t i e l y pr o po r ti o na l t o lI p 2 1 2

4 a nd a bs ol ut el y su ch t ha t 1l p & a m p c . a re e u al t o th e

alsfromoneanotherin order.

themeta lbetweenthecirclesK1 and

K 5 a n d K 6 K 7 a n d K 8 o n th e pa ra ll el s tr ai gh t

h , h 4 r e sp e ct i e l y. E n l a rg e t he s e ho l es

sothatthealteredstream-lines

, h 4 ( t h e s e po i nt s s up p os e d fi e d a nd v e r y

the irf lu -sheds. Whilea lw aysmainta ining

heholesandaltertheir positionsuntilthe

potentia lintheirl ipsbecome15 12 1 , 14

ntialbetweenthelips insuccession

. Inthuscontinuouslychangingtheho leswe

sarbitrarily butto f i ourideas w e

waysmadecircular.This ma esthe

e ceptthedistancef romthecircleHof the



s s_ u s e # p d



chmaybeany thingw eplease pro idedit

nto thediameterofthecylinder.

thusproposedisclearlypossible and

ue.Itis ofahighlytranscendental

aproblemformathematica lanaly sis butan

a landerror gi esitsso lutionbye lectric

uiteamoderateamountoflabourif moderate

eenf ina llyad ustedtofulf i lourconditions draw byaidof thevo ltmeterandmo ablee lectrodes the

abo ethegreatestpotentia lo f l ip1 andfor

ta lo f l ip4 andbetw eenthesee uipotentials w hichwesha llcallf andg draw n-1e uidifferent

hestream- lines ma inginf initesimal

ordingtotherulegi enabo einthe

oundthatthenumberofthe streamlinesism thesameasonour originalsurface andthewhole

uaresonthecylinderbetweenfandg

roughatfandg cutitopenbyany

andopenitoutf la t. Wethusha ea

yfourcur escuttingoneanotherat

dedintomninf initesimals uares correspondingindi idua lly tothemns uaresintow hichwedi ided

firstelectricprocess.Inthis chart

an scorrespondingtothe lips1 2 3 , 4

ere ise actcorrespondenceof the irf lu shedsandneighbouringstream- lines andof thedisturbances

ee uipotentials w iththeana logous

originalsurfaceascut forourprocess.

tricalproblemwasa necessityforthe

w hichIha ebeenoccupied andthisis

gitout thoughitm ightbeconsideredas

elf .



s s_ u s e # p d



N.

DUC TIONO F ELEC TR IC C UR R ENTS

GR A PHWIR ES.

ciationR eport 1855 P t. II. p. 22.

ys. Papers V o l. II. A rt. l x v . pp. 77 78.

R OU GHSUB MA R INEC A B LES ILLUSTR A TED

TEDTHR OU GHA MODELSU B MA R INE

E D B Y M I RR O R G A L V A N O M E TE R A ND B Y

ersinScotlandTrans. V o l. xv I. March18 187 , pp. 119

ys. Papers V o l. II. A rt. l x x v . pp. 168-172.

TR A TIONSO F THEMA GNETIC A NDTHE

R Y E F E C TS O F T R AN S PA R EN T B O D I ES

HT.

. V o l . v I I I . J u n e 12 1 8 56 p p . 15 0 -1 5 8 P h il . M ag .

5 7 p p . 1 9 8 â € ” 2 0 4.

Le c tu r es A p pe n di F p p . 56 9 -5 8 . i



s s_ u s e # p d



ND W A V E S I N A ST R ET C HE D U N I O R M C H A IN

RO STA TS .

Soc. P roc. V o l. v I. A pril8 1875 pp. 190-194.

rotating fly-wheel frictionlessly

eable f ramew or orcontainingcase. A

einwhichnot onlythefly-wheelbut

ymmetrica lroundthea iso f rotationof

ernategyrostatsandmasslessconnectinglin s andlettheconnection

r e j o i n t st a t e ac h c + 2

simplicity at

esof thegy ro- - - - -

ne linew hen

ight.This /Ci+

uil ibriuma is.

so f the

glin srespec- /

a n d X a n d /u t h e

thea iso f G-1

culartoit

o fagy ro- / i

landcase included

tof inertiao feachf ly -wheela lone roundits

tobetheangularv e locityoncegi ento

mainingalwaysthesamebecauseofthe

ots. Insteadof f irst in estigatinginf initesimalmotionsingeneral w eshallf irstta etheparticular

t limitedtobeinginfinitesimal.Then

iesimal bycomposit ionofcircular

en t s c f . Ro u th s A d a n ce d Ri g id D y na mi c s ~ 4 1 9. C f. a l so

aturalPhilosophy ~ 109.



s s_ u s e # p d



N

ntrarydirections andwithdifferent

tothe generalsolutionoftheproblemof

lengthofsuchachain tobeplaced

openplanepo lygon and theendsof

in g h el d f i e d b y un i e r sa l f le u r e j o i n t s l e t

otionperpendicularlytothisplanethat

esasarigidpolygonrotatingroundthe

w ithagi enangularv e locityn: re uired

andtheforcesonthef i edends sothat

tse lf maycontinuere o l inginthe

ingandgy rostaticlin s. . . Ci Gi C i+ l

idenotethe inclinationsofC iandGito

ds andlety ibethedistanceof thecentre

ne. Weha ethegeometricalre lation

4+ sinO i + c sin.+l......( 1 .

ro f theuni ersa lf le ure j o int( Thomson

e a ch g y ro s ta t ic l i n m o e s a s if i t s a i s w er e

ningthef i edends andthere j o inedtoa

ersa lf le ure j o int. Hencethe instantaneous

bisectstheanglew r-O ibetw eenthe line

o iningthef i edends. Itsangularv e locity

isis2n sin10i.The componentsofthis

ndinaplaneperpendiculartothea iso f

um-a is are

cos10i orn( 1-cos0i andnsin6i.

ntsofmomentumare

1 - co sO i a nd .. n si nO i .

to fmomentumofGi( caseandf lyw heel roundthea iso fGiis

cos oi + X ' w.

sn O i r o un d t he e u i li b ri u m a i s a nd

eplaneof thechain w eha e forw hole

omentumroundthelastmentionedline ,

1 - c os 0 i + X ' o s in 0 i -/ in s in 0 i co s 0i .

ouldbechangedhenceforth.



s s_ u s e # p d



H AI N O F G Y RO S T A T S

angularv elocitynina planeperpendiculartothee uilibriuma is andtheremustthereforebea

n ( 1 - co s0 i + X ' o s in i -F L n si n6 1c os O i ,

isperpendiculartotheplaneof the

thea iso fGi. Thedirectionof thiscouple

hastotendto increasetheangledi. We

wnthee uationsofmotion( ork inetic

hecomponentparalle lto thee uil ibrium

econnectinglin s mustbethesamefora ll.

onsofmotionparalle lto thee uil ibrium

eP: sothatPseciisthepull inthe

TheappliedforcesonGiarethepullso fCi

eso l ingthemweha e: Para lleltoe uil ibriuma is. Perpendiculartoe uil ibriuma is.

r i

P t an i + 1 l

ntreof inertiao fGi w eha ef ina lly

oe uil ibriuma is

i p e r p en d ic u la r t o e u i li b ri u m a i s

4 i + + t an A i . I g c os i

briuma isanddirectiontendingto increase

ofcentreof inertiao fGi

- t a n i = 0 . .. . .. . .. . .. ( 2 ,

( 1 - c os O i + X ' } s i n O i -/ Ln si n0 ic os O i

s di ( t an r s+ j + t a n i } . . . 3 ) .

2 , ( 3 ) , a p p li ed t o ea ch g yr os ta ti c li n , g i e a s

e ar e o f un n o wn q u a nt i ti e s % , 0 i y i i f

chaintobea gyrostaticandtheother

sothattherebethesamenumberof thetw o

sandinclinationsareinfinitelysmall

dif ferences asappliedbyLagrangeto



s s_ u s e # p d



N

nsofa" l inearsystemofbodies ( acaseof

becomesw hen w = 0 iscon eniently

1 , ( 2 , a n d ( 3 ) , w h en w e c an n e gl e ct t h e

, b e c om e

9 ~ + c.. 4 ,

' - ) = 0 . .. . .. . .. . .. . .. . .. 5 ,

P~ g IO - B i + ~ ( ' 3 r j + ~ r ) . ... ... .. 6 .

perationsuchthat

.. . . .. . . .. . . .. 7 ,

fi.

e

3

P g

a n d e l im i na t in g ' b e t we e n ( 4 a n d ( 5 , w e f i n d

p + P g( p ~ + 1 2 CP I yi = o

. .. 8 ;

course w ithO ior' 3 isubstitutedfory i.

ha etheq uadratic

0 . .. . .. . .. . .. . .. . . 9 ,

- A n2

2. .. . . .. . 10 .

o r V 1 1 e = s i n Ia . 1 .. .

ecomes

- 1

so lutionof (8 is

a . . . .. . . .. . . .. . . .. . 12 ;

ordinateof thecentreof inertiao fG

n a s z e r o , w e ha e

.. . .. . .. . .. . .. . ) .

+ B s in ~ - - - . . . . .. . 1 4 ;



s s_ u s e # p d



N S I N A C H AI N O F G YR O S T A TS 5 7

tesof inertiao f the lin slieonaheli

l e ng t h is

. . .. . . 15 ,

of particlesinthewa e-length1.

n........................... 16 ;

enotetheve locityofpropagationof the

w a emadeupbypropersuperposit ionof

e

/ a .. . .. . .. . .. . .. . .. . .. . 1 7 ;

,

. .. .. .. .. ( 1 8 ;

' n _ 2 P c /

w -tL 2 m

' n e- _ e n2

se pressionbecomeeache ualto

small thatistosay w henthew a e

nte lygreatincomparisonw iththedistance

e ighbouringmolecules andthee pression

.. . . .. . . .. . . .. . . .. . . ( 20 ,

ocityofpropagationofw a esinauniform

c be ingthemassperunito f length and

b u t v e r y sm a ll w e ha e a p pr o i m at e ly

_ _ 1 7 2 + c 2

77 - ( g )

e length. A ndby theappro imate

r V , o r n l/ 2 7r m n 2 = 4 - r 2 P ( g + c / 1 2 ap p ro i m at e ly . A ls o b e ca u se e a ch l i n i s v e r y s m al l i n al l i ts l i ne a r

nw ith1 p / ml2andX ' / m12areeachvery



s s_ u s e # p d



N

ablewithg2/12.Hencethesecond factorof

o imately

twofactors st i l lappro imate ly

I

+ c / l= 0 a pp ro i ma te ly . Th en

( g c

P g + c

m

5 1 8 84 .

tionnow consideredsupposes n tobev erygreat

4 1 8 8 ) .

R Y O F L I GH T .

eA cademyofMusic Philade lphia underthe

inInstitute Sept. 29 1884.

a n l i n In st it ut e V o l . LX X X V I II . No . 1 88 4 p p. 3 2 1 â € ” 3 4 1

i . 1 88 4 p p . 91 - 94 1 1 5- 1 18 .

uresandA ddresses V o l. i. pp. 300- 48.

A N D G R E E N S D O C T RI N E O F E X T R A NE O U S

NDYNA MIC A LLYF R ESNEL SK INEMATIC S

R AC TI O N .

. P ro c . V o l . x v . D e c. 5 1 8 87 p p . 21 - 3 ; P h il . M ag .

1 8 88 p p . 11 6 -1 2 8.

altimoreLectures pp.228-248.



s s_ u s e # p d



SISF O R ELEC TR O-MA GNETIC INDUC TION

RC U I T S W I T H C O N S EQ U E N T EQ U A T IO N S O F

IX EDHO MOGENEOU SSO LIDMA TTER .

ciationR eport 1888 pp. 567-570 Nature

p .5 6 9- 5 71 .

alformulastillneededforcalculation

uid motion w hichforbre ity Ica ll

Tertiary def inedasfo llow s: Half theve locity

mericallyanddirectionallywiththe

emolecularspin atthecorresponding

( short butcompletestatement the

ry istw icethespininthePrimary and

elocity intheTertiary isthespininthe

ertiarythemotionisessentially

andineachof themwenaturally

compressible f luidasthesubstance. The

rbitrarilyrestrict byta ingitsfluid

edtheproblem: Gi enthespinin

n to f indthemotion. Hisso lution

ntialsof threeidealdistributionsof

ingdensit iesrespecti e lye ua lto1/ 47r

nentsof thegi enspin and regarding

ialsasrectangularcomponentsofv elocity

n ta ingthespininthismotionasthe

edmotion. A pplyingthissolutionto f ind

ndary f romthev elocity inourTertiary

elocitycomponentsinourPrimaryare

ldistributionsof gra itationalmatter

rbre ity tosignify incompressible fluid



s s_ u s e # p d



N

especti e lye ua lto1/ 47rof thethree

ourTertiary.Thispropositionispro ed

5below bye pressingtheve locitycomponents

thoseofourSecondary andthoseof

hose ofourPrimary andtheneliminatingthev elocitycomponentsofSecondarysoastoha ethose

ofthoseofPrimary.

edso lidorso lidsofnomagnetic

ofelectricmotionin whichthereisno

andthereforenoincompleteelectric

hesame anycaseofe lectricmotionin

ectric currentagreeswiththedistributionofv elocityinacase ofli uidmotion.Letthiscase

numericallye ua lto47rtimesthee lectric

Tertiary.Thev elocityinourcorrespondingSecondaryisthen themagneticforceoftheelectric

heve locity inourPrimary isw hat

ca lledthe" e lectro -magneticmomentumatany

urrentsystem andtherateofdecrease

ycomponentof thislastv e locityatany

ingcomponentofe lectro -moti e force due

tionoftheelectriccurrent systemwhen

nge. Thise lectro -moti e force combined

e if there isany constitutesthew hole

anypo into f thesystem. HencebyOhm s

lectriccurrentatanypo intise ua lto

multipliedintothesumofthe correspondingcomponentofelectrostaticforceandtherateof decrease

espondingcomponentofv elocityof

s ym bo ls l et ( u 1 v l w I , ( u , v W , w 2 ,

w ) d e no t e re c ta n gu l ar c o mp o ne n ts o f t he v e l oc i ty a t

, y z ) o f ou r P ri m ar y S e co n da r y a n d Te r ti a ry .

d - d1 ul

' d dy. ... .. 1 ,

e ll- now nelementary theoremV 2V = -47rp.

n et i sm ~ 5 1 7 ( p o s t sc r i pt ( c .

eism ~ 604.



s s_ u s e # p d



U A T I O N S F O R E L E CT R IC P RO P A G A TI O N 5 4 1

d 2 du .

= â € ” - -- W 3 = ...... 2 .

d ' d dy

w 2 fr om ( 2 b y ( 1 , w e fi nd

d l 2 d u ..

) 2 _ ( ^ d t -+ d y2 + t d 2 ) ^ } & a mp c .... 3 ) .

~ 2 o f incompressibility inthePrimary

s

v 3 = - V 2 V 1 w = - V 2 w1 .. .. .. .. . 5 ,

v i i . ( N o e m be r 1 8 46 o f m y Co l le c te d

a lPapers( V o l. I. ,

. .. . . .. . . .. ( 6 .

6

edproofof ~ 3 .

denotethecomponentsofe lectriccurrent

n t h e e le c tr i c sy s te m of ~ 4 s o t ha t

4 T r y= v 3 = - V 2 l 4 rw = W 3 = - V 2 w .. . 7 ,

4 , g i e

electro-moti eforcedueto changeof

4 V 7 r V - 2 d .. - 2. . ( 9 ,

dt

electrostaticpotentia l w eha e forthe

cmotion( . 4

dd

-1 ,

ofthespecificresistance.

a te r ni o ni c r ea s on s t a e s V 2 t h e n eg a ti e o f m in e .



s s_ u s e # p d



N

accordingto~ 4 w emay

ent put

/ + = . .. ... ( 1 ,

x cd

a le nt s to ( 1 0 ,

K ; d t 2 ) . .... 12 .

iminationofT maybeillustrated

ample a f initeportionofhomogeneoussolid

e( a longthinwirew ithtw oends ora

s ol i d gl o be o r a l um p o f a ny s h ap e o f

ogeneousthroughout , withaconstant

edthroughitby electrodesfroma

rsourceofe lectricenergy andw ithproper

o leboundary soregulatedastok eepany

aate erypo into f theboundary w hile

late throughtheinteriorbyv arying

riortoit.There beingnochanging

posit ionof~ 4 Pcanha enocontributionf romelectrif icationw ithinourconductor andtherefore

. .. . . . 1 ) ,

nd ( 1 1 , g i e s

.. . . ( 14

( 1 4 w e h a e f o u r e u a ti o ns f o r th r ee u n n o wn

ecaseofhomogeneousness( cconstant are

ree because inthiscase(14 fo llow sf rom

4 issatisf iedinit ia lly andthepropersurface

pre entanyv io lationof itf rom

ntthroughoutourf ie ld thefoure uations

remutua lly inconsistent f romw hichitfollows

nchangingnessofe lectrif ication(~ 4 isnot

ngandimportantpracticalconclusion

are inducedinanyw ay inaso lidcomposed

te lectricconducti it ies( piecesofcopper

pe f i e d t og e th e r in m e ta l li c c on t ac t t h er e



s s_ u s e # p d



U A T I O N S F O R E L E CT R IC P RO P A G A TI O N 5 4

inge lectrif icationo ere ery interface

conclusionwasnotat firstob iousto

obyanyoneapproachingthesub ect

mathematicalformulas.

heterogeneousnessuntilwecome

rificationandincompletecircuits letus

ntehomogeneousso lid. A s( 8 ho ldsthrough

rsupposit ionin~ 4 andasK isconstant

dthroughallspace andtherefore = 0 w hich

dw

- â € ” ; = _ -.. . 1 5 .

d t

esssimplythek nownlawofelectromagneticinduction.Ma well se uations( 7 of~ 78 ofhis

becomeinthiscase

2 & a m p c .. .. .. .. .. .. . 1 5 ) ,

thin , accordingtoanyconce i able

ctricconducti ity w hetherofmeta ls or

r es i ns o r w a , o r s he l la c o r i nd i a- r ub b er o r

s orso lidorli u ide lectro lytes be ing as

dforcompletecircuitsby thecuriousand

to menotwhollytenablehypothesis

610 , forincompletecircuits.

uggest forincompletecircuits

y inge lectrif ication issimply thatthe

-moti eforcedueto electro-magnetic

du/ dt & amp c. Thus forthee uationsof

p ly t o k e e p e u a ti o ns ( 1 0 u n ch a ng e d w h il e

b ut i ns te ad o fi t ta i ng ' V / 2 du d _ dw = V d d p

d = d t. .. .d p ( 1 6 ,

d t

the electricdensityattimet andplace

a n d ' v ' d e no t es t h e nu m be r o f el e ct r os t at i c un i ts i n t he

efundamenta lpostulatethatrateofchangeofe lectricdisplacementoperatesascurrent andsoma esa llcurrentsf low ef fecti e ly incomplete

nowonlyofhistoricalinterest representsprobably

fMa w ell sscheme promptedbyHert ' sthenrecent

a es.



s s_ u s e # p d



N

ectricq uantity.Thise uatione pressesthattheelectrificationofwhichT isthepotential

nanyplace accordingaselectricity

more inthanout. Wethusha efour

n d( 1 6 , f or ou rf ou ru n n o wn s u v , w P a nd

olutions withnothingv agueordifficult

l ie ew henunderstood by the irapplication

rtoconce i able idea lproblems suchas

ryortelephonicsignalsalong submarine

andlines electricoscillationsina finite

form transferenceofelectricitythrough

p c . & a m p c . T hi s h o we e r d o es n o t pr o e m y

tisre uiredforinformingusastothe

ctsofincompletecircuits and as

ed it isnoteasy to imagineanyk indof

decidebetweendifferenthypotheses

netry ingtoe o l eoutofhisinner

hemutualforce andinductionbetween



s s_ u s e # p d



R ENC EOF ELEC TR IC ITYWITHINA

L I D CO N D U C T O R .

c i at i on R ep o rt 1 8 88 p p . 57 0 5 7 1 N a tu r e

.5 71 .

onandformulasofmypre iouspaper

, a n d ta i n g p to d e no t e 47 r t im e s th e e le c tr i c

p ac e ( x , y z ) , w e ha e

2f du d dwA.*

a + d t. . .. . .. . .1

y d j

, w T b y t hi s f ro m ( 1 0 , w e f i n d o n t h e

t

. . . .. . . .. . . .. . . .. . 18 .

ryconditions whenafinitepieceof

b e c t i n o l e s c on s id e ra t io n o f it v , w

u a ti o ns ( 1 7 a n d ( 1 2 m u st b e t a e n i nt o

esub ectisaninf initehomogeneoussolid

w enow suppose ittobe ( 18 suf fices. It is.

remar thatthisagreesw iththee uation

ouse lasticf luid foundfromSto es s

irw ithv iscosity ta enintoaccount and

, w g i e n by ( 1 7 a nd ( 1 0 , w he n p ha s

ewiththev elocitycomponentsofthe

ndnaturalenoughsuppositionbemade

actsonlyaga instchangeofshape andnot

mewithoutchangeofshape.

me

s . .. . .. . .. 1 9 ,



s s_ u s e # p d



R OPA GA TIO N [ 68-70

utionin( 18 ,

. .. . .. . .. . .. . .. . .. . . 2 0 ,

* * 1 20. * * * * * * * * * * * * ( 2 0 ,

r2 + ) + ( . . . .. . . .. . . .. . . .. . 21 .

h e q u a dr a ti c ( 2 0 f o r q ,

.. .. .. .. .. . 2 2 .

totheSectionnumericalillustrations

cillatorydischargeweregi en.

N S O F F O U R I ER S L AW O F D I F U S IO N

A G RA M O F C U R V E S W I T H A B S O L U T E

S.

c i at i on R ep o rt 1 8 88 p p . 57 1 -5 7 4 N a tu r e V o l . x x x v I I I.

y s. Papers V o l. III. A rt. x c ii i . pp. 428-4 5.

LIGHTNINGC ONDUC TO R SA TTHE

N.

c i at i on R ep o rt 1 8 88 p p . 60 - 6 06 N a tu r e V o l . x x x v I I I.

46 .



s s_ u s e # p d



O N A N D R E R A CT I O N O F L I GH T .

o l . x x v I . p p. 4 1 4- 4 25 N o . 1 8 88 a n d pp . 5 00 5 0 1

a l ti m or e L ec t ur e s p p . 17 4 3 5 1 - 5 4 4 0 7.

TY A NDPONDER A B LEMA TTER.

ne e rs J o u rn a l V o l . x v I I I . 18 9 0 p p . 4- 7 ( I n au g ur a l

1 88 9 .

s.Papers V ol.III.Art.cii. pp.484-515.

M F O R T HE C O N S T IT U T I O N O F E T HE R .

oc . P ro c . V o l . x v I I . Ma r ch 1 7 1 8 90 p p . 12 7 -1 2 .

ys. Papers V o l. III. A rt. c. pp. 466-472.

I SC O U S L I Q U I D E Q U I L I B R IU M O R M O T IO N

D E Q U I L I B R IU M O R M O T IO N O F A N

A LLEDF O R B R EV ITY" ETHER ; MEC HA NIC ALR EPR ESENTATIONO F MA GNETIC F O R C E.

dPhys. Papers V o l. III. A rt. x ci . May 1890



s s_ u s e # p d



7 5

IMENTSF O R C OMPA R INGTHEDISC HA R GE

H RO U G H D I F E R EN T B R A N CH E S O F A

LOR DK ELV INandA LEX A NDER

ciationR eport 1894 pp. 555 556.

emetallicpartofthe dischargechannel

wo linesofconductingmeta l eachconsistinginparto fatest-w ire theotherpartso f thetw olines

ape materia l andneighbourhood o f

pectto facilityofdischargethrough

.

asnearlyasweha ebeenhitherto

a landsimilar andsimilarlymounted. Each

tinumwire of' 006cm.diameterand

etchedstraightbetweentwometalterminals

.O neendofthe platinumwirewas

assmounting theotherw asf i edtoa

armformultiplyingthemotion.The

de elopedinthetest-wirebythe

itse longation theamountofw hichw as

traced by theendof themultiply ingarm

a mo ingcylinder. TwoofLord

e lectrostaticv o ltmeters suitablerespecti e ly for

000and1 500 w erek eptconstantlyw ith

theouter coatingsoftheleyden and

heinside coatingsoftheleyden.

thertomadethetwow irestobe

eenofthesamelength. When theywere

utofdif ferentdiameters thetesting

w astobee pected thatthetest-w ire in

hic erwirewasmoreheatedthan the



s s_ u s e # p d



EDISC HAR GEINDIV IDEDC IR CU IT

h.In acontinuationofthee perimentswehopetocomparehollowandtubularwires ofthesame

ndsamelengthandsamemateria l.

non-magneticmateria l- fo re ample copperandplatino id-o f thesamelength butofvery

astoha ethesameresistances thetesting

earlye ua l.

rimentsthetestedconductorswere

each' 16cm. diameter 9metreslong and

w h ic h i t w il l b e ob s er e d i s v e r y s ma l l in

msin eachoftheplatinumtest-wires.

w asco iledinauniformheli o f forty

m. diameter.Thelengthoftheheli

edistancefromcentretocentreofne ighbouring

middle oftheothercopperwirewas

mtheceil ing andthetwohal espassed

epointsofj unctioninthecircuit.

wireinthis channelwasmorethan

hetest-wireinthe channelofwhich

se enty-onevarnishedpiecesofstra ight

within theglasstube whichwasas

Thismadethetestingelongationten

rchannel.

w hichw eha emadehasbeenbetw een

conductors.The lengthofeachwas

erof the ironw irew as' 0 4cm. andits

ms. Thediameterof theplatino idw irew as

sstance6 82ohms. Eachof thesew ireswas

eadf romtheceil ing attachedto itsmiddle

neof thetestedconductors . F ourteen

e se enw iththetest-wiresinterchanged

sin whichtheywereplacedforthefirst

leshowsthemeansof theresultsthus

gardingtheelectrostaticcapacitiesof the

o ltagesconcernedintheresults.

ars connectedtoma ev irtuallyone

rofarad w erechargedupto9 000vo lts and



s s_ u s e # p d



N

dedchannel. Theenergy therefore in

ewas11105 x 106ergs.Ineach ofthe

o ltsw erefoundremaininginthe j arsaf ter

he lastfour1 400.

incms.

n Energyused

channelcontain-Inchannel containingplatinoid ingiron

2 x 1 0 6 er g s - 0 17 9 4 0 1 22 6 , ) ~ , , - 0 18 6 1 01 8 29 - 0 1 2 47 - 01 2 9

4

4x 106ergs -0182 ) -01276

01828 01244

eelongationofthe test-wireswas

mtheprecedingdescription somewhat

ngtoseethatthemeanresultsinthe

84megalergsofenergyusedaresonearly

etw ocircuitsare inthetw ocases

d1-46. Theconclusionthattheheatingef fect

iththeplatinoidwireisnearly one-anda-halftimesasgreatasthatofthe test-wireinserieswiththe

g notonly initse lf butinre lationto

e se ceedingly interestingandinstructi e

ai epathsforthedischargeof leyden- ars

nLightningConductorsandLightning

renotdecisi e inshow inganygenera lsuperiority

hesamesteadyohmicresistance bute en

emingsuperiorityof theironforefficiency

en- ar. Ourresult isq uitesuchasmight

rome perimentsmadeeightyearsagoby

edinhis paper" O ntheSelf-induction

undConductors .

I I . 18 8 6 p . 4 69 .



s s_ u s e # p d



5 51 )

R YO F R E R A C TION DISPER SION

ISPERSION.

c i at i on R ep o rt 1 8 98 p p . 78 2 7 8 ; N a tu r e V o l . LV I I I .

46 5 47 .

altimoreLectures p.148.

N D U L A T O R Y T H EO R Y O F C O N D EN S AT I O N A LR A RE A C TI O N A L W A V E S I N GA S ES L I Q U I D S A N D SO L I D S

L W A V E S I N S O L I DS O F E L EC T RI C W A V E S

C A PAB LEO F TR A NSMITT INGTHEM A ND

V I S IB L E L I G HT U L T R A -V I O L E T L I GH T .

c i at i on R ep o rt 1 8 98 p p . 78 - 7 87 N a tu r e V o l . L I X .

. 56 5 7 P h il . M Ia g . V o l . X L V I . N o . 1 8 98 p p . 49 4 -5 0 0.

ectures pp. 148-162.

O N A N D RE R A CT I O N O F S O L I T A RY P L AN E

N TE R A C E B E T W E EN T W O I S O T R O P I C

I D S O L I D O R E T H E R.

c . Pr o c. V o l . x x I I . De c . 19 1 8 98 p p . 3 6 6 - 7 8 P h il .

eb. 1899 pp. 179-191.

a lt imoreLectures ~ ~ 112-121.

SELLMEIER SDYNA MIC A LTHEOR YTO THE

R O D U C E D B Y S O D I U M - V A P O U R .

c . Pr o c. V o l . x x I I . F e b . 6 1 8 9 9 p p . 5 2 - 5 1 P h il .

r ch 1 8 99 p p . 3 0 2 - 0 8 .




s s_ u s e # p d



RO P A G A TI O N [ 8 0 8 1

NO F F O R C EWITHINA LIMITEDSPA CE

D U C E S PH E RI C AL S O L I T AR Y W A V E S O R

CW AV ES O F B O THSPECIES EQ U IV O LU MINALANDIRRO TATIO NAL INANELASTICSO LID.

o l . X L V I I . M ay 1 8 99 p p . 48 0 -4 9 ; V o l . X L V I I I . A ug u st

O c t . 1 8 99 p p . 3 8 8 - 9 ; a l so r e ad a s a P re s id e nt i al

a thematica lSociety cf . V o l. xx x I. J une8 1899


PR O DUC EDINA NIN INITEELA STIC SOLID

R U G H T HE S P AC E O C C U P I ED B Y I T O F A

N L Y B Y A T TR A CT I O N O R R EP U L S I O N .

. P ro c . V o l . x x I I I .J u l y 16 1 9 00 p p . 21 8 -2 5

1900 pp. 181-198 C ongresInternationa lede

st io n d e 19 0 0 V o l . II . p p. 1 - 22 .

ectures A ppendi A pp. 468-485.



s s_ u s e # p d



ETHER F O R ELEC TR IC ITYA ND

o l . L. S e pt . 1 90 0 p p . 3 0 5 - 0 7 .

shedinthe lastnumberof the

ofw hichthisisacontinuation Il im ited

hematicaldynamics andmerely

ffindingin itane planationofthe

heU ndulatoryTheoryofLightreferred

agraphs( ~~ 1 18 . Thefo llow ing

tanceofasupplementarystatement

enora lly totheC ongresInternational dePhysi ueatameetingheldinParislastWednesday

temptationtospea ofefforts

roperassumptionsforincludingsomethingofthealliedsub ectsmentionedinthefootnoteon~ 1.

icity w hich fo llow ingLarmor Ia t

it ine itablyoccurstosuggestaspecia l

theconditionstatedinlines 12-22

twouldbeanatom whichbyattraction

paceoccupiedby itsv o lume anda

beanatomwhich by repulsion rarefies

spaceoccupiedbyitsv olume.The

r outsidetwosuchatomsbythe

hichtheye ertontheether within

arentattractionbetw eenaposit i eanda

ngof thesectionsiscontinuousw iththato fNo. 81.

this maybetheresinouselectrification butitmay

.Itmustberememberedthatv itreouselectrificationhas

i emere lybecause it isitw hichisgi enby the" prime

rdinaryelectricmachine.



s s_ u s e # p d



N

dapparentrepulsionbetweentwoelectrons

egati e .

ttractionsandrepulsionswould

iminisheddistancethanaccordingto

in erses uare . Thislaw w hichwe

ndC a endishtobetruefore lectric

cannotbee pla inedbystressinether

orhithertoimaginedpropertiesofelastic

mplehypothesis assumingactionatdistances

sofether e plainsitperfectly.Consider

py inginf initesimalv o lumesV , V ' , a t

pothesisisthat theyrepelmutually

V '

. . .. .. .. .. .. .. .. .. ( 1 ;

hedensit iesof thetw oportionsofether

naturaldensityofundisturbedether.

ulsionor attractionaccordingas( p-1 ,

ameorofoppositesigns andz ero ife ither

ansthatetherofundisturbednatura ldensity

actionnorrepulsionfromanyotherportion

blesA epinus doctrineof themiddle

commonly referredtoasthe" onef luidtheoryofe lectricity butnow insteadofe lectricf luid w e

elasticso lidper adinga llspace. A ccordingto

similarelectricatomsrepeloneanother

inv irtueof forcebetw eeneachatomand

t andmutualrepulsionorattraction

ith nocontributi eactionofthe ether

andbetweenthem.

ngthusfreedfromtheimpossibletas

rostaticandmagneticforce is( w emay

ompetenttoperformthesimplerdutyof

cealone.

yinsuperableobstacleagainst.

racticalreali ationhasbeenthe

manywellk nowncasesofmagnetic



s s_ u s e # p d



RY

les whetherdueto steelmagnetsor

ringthat inourmostdelicatee perimentsinv ariousbranchesofscience ponderablebodieslargeand

emo edf reelyby forcesof lessthana

ness o famill igram how canw econce i e

eymo etobecapableof thestress

ssionofforce betweenflatpolesofan

gpers uarecentimetretomorethantwo

inessofak ilogram Thisdiff iculty is

pothesiswhichI ha edescribedto

o e . Wemaynow supposethedensityof

se sub ectonly tothe lim itationthatit

disturbsensibly theproportionalityof

ity indif ferentk indsofmatter pro edby

p e ri m en t f o r le a d b r as s g l as s & a m p c .

fK epler sthirdlaw forthedif ferent

bablywemightsafely ifwewishedit

rto beasmuchas 10-6. Iamcontent

tosuggest10-9. This w iththeve locityof

etrespersecond ma estherigidity ( be ing

e l o c it y e u a l to 9 . 10 1 d yn e s pe r s u a re

mewhatgreaterthantherigidity ofsteel

lynotforw antofstrengththatweneed

ceofethertotransmitmagneticforce

opefulo f see ingso l edsomeof the

sw hichmeete eryef forttoe pla in

duction andelectromagneticforce and

eelmagnet bydefinitemechanicalaction

guityusethesimpleword" w eight here becausethis

s andispracticallyusedmoreoften tosignifyamass.

ea inessofamass.

eticf ie ldhithertomeasuredis Ibe lie e thato fDubo is

netismtothisCongress[ seeunderNo. 81 inw hichhe

weentwosmallplaneend-facesofsoftiron polesofa

Thisma estheattractionpers uarecentimetreof

2 - 87 r o r a pp r o i m at e ly 2 . 1 07 d y ne s o r 2 0 k i l o g r am s .



s s_ u s e # p d



Y INGMETHODF O R STR ESSA NDSTR AIN

. P ro c . V o l . x x I . J a n . 20 1 9 02 p p . 97 - 10 1

. 1902 pp. 95-97 A pril 1902 pp. 444-448.

stressandstrain hithertofollowed

hasthegreatdisad antagethatit

stra in tobe inf initely small. A sa

eincon eniencethatthespecifying

tiallydifferentk inds( inthenotation

f g s i mp l e el o ng a ti o ns a b c s h ea r in g s .

o idedifw eta ethesi lengthsof the

onof theso lid orw hatamountstothe

le thethreepa irso f face-diagona lso fa

especify ingelements. ThisIha ethought

s butnotti l lto -day ( Dec. 16 ha e

n enientlypracticable especia lly for

genera li eddynamicsofacrysta l.

olid tobeahomogeneouscrystalof

utfromita tetrahedronAB CD ofany

thethreenon-intersectingpairs( AB ,

D , ( C A B D o fi ts si e dg es be de no te db y

( 3 q , 3 q ' ) , ( 3 r 3 r ) . ............. 1 .

, q ' ) , ( , . .. .. .. .. .. .. .. .. . 2

trahedron sim ilartoA B C D formedby

a 3 , y 8 t h e ce n tr e s of g r a i t yt o f t he f o ur

ngaf igureboundedby threepa irso fparalle lplanes is

hy butthe longerandlesse pressi e para lle lepiped is

eadofit bymathematicalwritersandteachers.Ahe ahedronwithitsanglesacuteandobtuse iswhatiscommonlycalled bothinpure

ography arhombohedron.Aright-angledhe ahedronis

Gree orotherlearnednameishithertotothefrontinusage

a lhe ahedronisacube.

enceforthca llthecentreofgra ityofatriangle oro f

scentre.



s s_ u s e # p d



IC A T IO NO F STR ESSA NDSTR AIN

C D A D A B , A B C r e s pe c ti e l y s o t ha t we

= , 8 y r = y c/ p = y 8 q ' = a S r = / . Con si de rn ow

untsofwor doneby thesi pa irso fba lancing

stress-componentsdescribedin~ 2

ntsv ary fore ample theba lancing

w hena/ increasesf romptop+ dp. a ll

, r p , q ' , r r e ma i ni n g co n st a nt . F o r t h e

maysupposetheopposite forces P tobe

steadofbe inge uablydistributedo erthe

cethew or w hichtheydo isPdp and

ci ng p ul ls Q , R P , Q ' , R , d o no w or .

apply tothefacesA DC B DC e ual

e ua llydistributedo erthem. Thesetwo

astressora stress-component.

achof thef i eotheredgesapplyba lancing

uttingit. Thusw eha eina llsi

eltothesi edgesofthetetrahedron

Q ' ) , ( R R ) . ... ... ... ... .. 3 ) ;

forces appliedastheyareto the

areba lancedinv irtueof themutua lforces

henitsedgesareof thelengthsspecified

o , q o q 0 , r o r o , b et he v a lu es of th es pe ci fy in g

hennoforcesareappliedtothefaces. Thus

eva lues o f thesi lengthsshownin

presentthestra inof thesubstancew henunderthe

) .

ew henpullsuponthefaces each

aregradua lly increasedtotheva luesshown

e of t h is p r oc e ss w e ha e

Q d + Q ' d ' + Rdr + R d r . .. 4 .

w e p r es s ed a s a f un c ti o n of p p , q , q ' ,

w = dw dw R

' . , d = r dR

d d r d . .. .. . .. . 4 .



s s_ u s e # p d



N

tionofthemolardynamicsof an

enera lpossiblek indaccordingtoGreen s

dintermsof thenew modeofspecify ingstresses

lythestateof strainspecifiedby

t th e t et r ah e dr o n of r e fe r en c e A o B o C oD o f o r th e

andstress bee uila tera l( thatistosay

of~ 2 ( 1 le tIo feachedge

o = r o .

sphericalsurfacetouchingeachof thesi

eatK0 thecentreof thetetrahedron

mustbethemiddlepointsof theedges.

eneousstra in , to thecondition( p q , r

i n w hi c h Ao B o C o Do b e co m es A B C D . T he i n sc r ib e d

sanell ipsoidha ingitscentreatK the

touchingitssi edgesatthe irm iddle

sfully andclearlythestateof strain

p , q ' , r . I t is w ha t is c al le d th e " s t ra in

ellipsoidtouchingthesi edges

ious. (1 ThroughA B andC Ddraw

ralle ltoC DandA B anddealsim ilarly

fnon-intersectingedges.Thethree

sfound constituteahe ahedronw hich

lipsoidtouchingthesi f acesatthe ir

wA , B K , C , D , a nd pr od uc et oe u al

' , K C , K D b ey on d K . W e t hu s fi nd f ou r

C , D , w hi ch w it hA B , C D a re th ee ig ht co rn er s

chw efoundbyconstruction( 1 . A circumscribedhe ahedronbeingthusgi en theprincipa la esof the

ntation arefoundby theso lutionofacubic

he strain-ellipsoid whichisin

andw hichhasthead antagethatinits

' N a tu r al P h il o so p hy ' ~ 1 5 5 ' E l em e nt s ' ~ 1 6 .

stingtheoreminthe geometryofthetetrahedron:Ifanellipsoidtouchingtheedgesof atetrahedronhasitscentre atthecentreof

ntsofcontactare atthemiddlesofthe edges.

' N a tu r al P h il o so p hy ' ~ 1 0 0 ' E l em e nt s ' ~ 1 4 1 .



s s_ u s e # p d



IC A T IO NO F STR ESSA NDSTRA IN

eusoutsidethe boundaryofour

isasfo llow s: -Inthee uila tera ltetrahedronA oB o C oDodescribe f romitscentreK0 aspherica lsurface

aces.Ittouchesthesefacesat their

chesthefourthface andatitscentre .

edeterminate one-so lutiona l problemto

attheir centresanythreeofthefour

B C D andha ingitscentreatK, this

re thefourthfaceofthe tetrahedron

dforthehomogeneousstrainbywhich

onofso lidisa lteredtothef igureA B C D.

odofspecifyingstrainandstress

arymethodfor infinitesimalstrains

sses:-LetX denotethelengthofeach

tetrahedronof re ference A oB oC oDo and

ubeofw hichA 0 B 0 C o Doarefour

gthehe ahedronfoundbyapplyinge ither

5tothetetrahedronA oB oC oDo . The

f thiscubeareeache ua ltoX , andtherefore

cubebeinfinitesimallystrainedsobhatits

, h ( 1 + f , h ( 1 + g ; a nd s ot ha t th e an gl es

realteredfromright anglestoacute

ngrespecti e lybya b cf romright

e f g a b c i n t he n o ta t io n o f

toin theintroductoryparagraphabo e.

metryoftheaffair weeasilyfindthe

oftheface-diagonals whichaccording

ar e ( p - 1 X , ( p - 1 X , ( q - 1 X , e tc . a n d

:

b

- b . . .. . .. . .. . .. . .. .. . .. 5

etwospecificationsof anyinfinitesimal

nddenotinge+ f+ gbys w ef ind

' + , r - 6 = 2s .. .. .. .. .. .. .. . 6 .



s s_ u s e # p d



N

c e f g i nt er ms of p q , r p , q ' , r , w e ha e

- ' ; c = r -r ; l

s - - ' + 2 g= s -r -r + 2 ( 7 .

edtoproduceaninf initesimalstra in

c i n a h om o ge n eo u s so l id o f c ub i c cr y st a ll i ne

bythefollowingformula:

g 2 + 2 3 ( f g+ g e+ e f + n ( a 2 + b 2+ c 2 . . .. 8 .

lymodifiedbyputting

n = I - A ( - . ) . .. .. .. .. .. . 9 ,

ul modulusandn , nthetw origiditymoduluses. Withthisnotation( 8 becomes

g 2 + 2 n [ ( f -g 2 + ( g - e 2 + ( e - f 2

. .. .. . 1 0 .

earingsparallelto thepairsofplanes

chisthesamething changesof theanglesof

efacesfromrightanglesto acuteorobtuse

gidity re lati e tochangesof theangles

hefacesfromright anglestoacute

compressibilitymodulusisk .U sing

w e ha e

q + q ' - -r ) 2 + ( r + r -p -p ) 2

) 2 + n ( p _- p 2 + ( q _ q ) 2 + ( r - r 2 ... 11 .

ETHER EA LTHEOR YO F THEV ELO C ITY

L I Q U I D S A N D SO L I D S .

c i at i on R ep o rt 1 9 0 , p . 5 5 P h il . M /a g . V o l . v I . O c t . 1 9 0 ,

ectures x x . pp. 46 -467.



s s_ u s e # p d



b it s 5 1 ; r e du c ed t o

5 1

p r es s ur e d ue t o 1 88

on98

ccompany ing282

es77

eadingw a eson

onstancy50

470

ing464 gy rostatic475

n52

place s intida l

, 5 2 1

, motion steady through f ree

0 1 , p o te n ti a ls 4 1 s e . 5 6

aperson551

propagationin3 0

3 0

odifiedforcyclic

ai e f luidpressure

esshedof f217

fmotion472

andstrain

hofdisrupti e548

tioninopen

ctor545

onoffluidin 19

abil ityofv ortices

ples174

ar

minimum181 to li uidgy rostat

cularpartit ion484

a i m um f o r gi e n i mp u ls e

g e n

ron547 duties

ortices stabil ity

ids97 dueto

dicfunctions35

ip p le s 9 1

ifuga l460 pendulum

w a e s 3 0 4 4 0 1

s ta bi li ty 1 3 , 1 8 ;

eriments482

y rostats5 3

2 o f c ha i n 5 3

motionin211

n determinate

ton o f2 ;

27 , incyclicmotion6



s s_ u s e # p d



luesof appropria te

91

cmotion

trigonometry51

s 4 6

ousf luid disturbed3 3 0 , motion R eyno lds criterion

e - mo t io n i n tu r bu l en t

h eo r y of 3 1 7

eticana logy94 99

of f loating1 5

yof474

n g e ne r al i e d 5 2 ,

gy484 495

mination516 521

n s ta b il i ty o f v i s c ou s

al c l oc 4 6 , 4 7 0

490 stabil ity515

o n r ip p le s 9 1

2

fv iscousf low 33 0

toanobli ue lymo ingplane

3 6

onthecriterionof

1 3 3 5

mwa es79

nimumv elocity91

pass464

id f igureof189 , w ater gra itationa loscil la tionsof141

uid54

n wi n d an d wa e s 8 1

a tt e rn 4 1

3 3 6

7

negati e

mo t io n o f i n l i u i d

to n 72 , i n m o i n g f l ui d f o rc e s on 9 ;

4

cyclic

insimple

2

a itatingf luid

dperiodicf luid

menta l

iscousf low betw eentwo

itha

0

agationinturbulentmedium308

79

lsin5 2

partition484

v orte of free

269 Airy s

eory

basin

54 , in inlandsea ef fecto fearth s

e s



s s_ u s e # p d



e fl u id s l ip s 1 87 , o f f re e mo b il i ty 9 7 , r i ng f l ui d c on e c te d b y 7 , r i ng i n er t ia o f 9 e n er g y

rings interactiononco ll ision

f fl o w of 6 , , r i ng t r an s la t or y m ot i on o f 6

suredueto impact

lord kelvin volume 4

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