perry, john - the calculus for engineers

8/14/2019 Perry, John - The Calculus for Engineers

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CALCULUS FOR ENGINEERS.


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P R I N T E D B Y J . & C . K . C L A Y ,A T T H E U N I V E R S I T Y P R E S S .


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T H E

CALCULUS FOR ENGINEERS

J O H N P E R R Y , M.E., D.SC, F.R.S.fill. SCN., ASSOC. ME JIB. INST. C.E.,PROFESSOR OF MECHANICS AND MATHEMATICS IN THE BOYAL COLLEGE OP SCIENCE,

LONDON; PRESIDENT OF THE INSTITUTION OF ELECTRICAL ESOINKEBS.

FOURTH EDITION. REVISED.

LONDON:E D W A R D A R N O L D ,

37, BEDFORD STREET.

[All lliyhts reserved."]

B Y


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P R E F A C E .

T H I S book describes what has for many years been themost important part of the regular course in the Calculusfor Mechanical and Electrical Engineering students at theFinsb ury Tech nical Coll ege. It was suppl ement ed by easywork invol ving Four ier, Spheric al Harmo nic, and Besse lFunctions which I have been afraid to describe here becausethe book is already much larger than I thought it wouldbecome.

Th e stu den ts in Octo ber kne w only the most ele ment arymathe matic s, ma ny of the m did not know t he Bin omi alTheor em, or th e defin ition of the sine of an angle. In J ul ythe y had not onl y done the work of t his book, bu t the irknowle dge was of a practi cal kin d, ready for use in anysuch engineering problems as I give here.

One such student , Mr No rma n E ndaco tt, has correctedthe manu scri pt and proofs. H e has wor ked out man y ofthe exerci ses in the thi rd chap ter twic e over. I th an k himhere for the care he has ta ke n, and I ta ke leav e also tosay tha t a syste m which has, yea r by year, produ ced man ymen wi th his kin d of kno wl ed ge of mat hem at ic s has agood deal to rec omm end it. I say thi s th rou gh no vani tybut becaus e I wish to encou rage the earn est stud ent. Bes ide sI cannot claim more than a portio n of the credit , for Ido not th in k th at the re e ver before was such a co mpl ete


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v i PREFACE.h a r m o n y i n t h e w o r k i n g o f a l l t h e d e p a r t m e n t s o f a ne d u c a t i o n a l i n s t i t u t i o n i n l e c t u r e s a n d i n t u t o r i a l , l a b o r a t o r y , d r a w i n g o f fi ce a n d o t h e r p r a c t i c a l w o r k a s e x i s t s i nt h e F i n s b m y T e c h n i c a l C o l l e g e , a l l t e n d i n g to t h e s a m ee n d ; t o g i v e a n e n g i n e e r s u c h a p e r f e c t a c q u a i n t a n c e w i t hh i s m e n t a l t o o l s t h a t h e a c t u a l l y u s e s t h e s e t o o l s i n h i sb u s i n e s s .

P r o f e s s o r W i l l i s h a s b e e n k i n d e n o u g h t o r e a d t h r o u g ht h e p r o o f s a n d I t h e r e f o r e f e e l d o u b l y s u r e t h a t n o i m p o r t a n tm i s t a k e h a s b e e n m a d e a n y w h e r e .

A n e x p e r i e n c e d fr ie n d t h i n k s t h a t I m i g h t w i t h a d v a n t a g eh a v e g i v e n m a n y m o r e i l l u s t r a t i o n s o f t h e u s e o f s q u a r e dp a p e r j u s t a t t h e b e g i n n i n g . T h i s is q u i t e p o s s i b l e , b u t i fa s t u d e n t f o l l o w s m y i n s t r u c t i o n s h e w i l l fu r n i s h a l l t h i s s o r to f i l l u s t r a t i o n v e r y m u c h b e t t e r fo r h i m s e lf . A g a i n I m i g h th a v e i n s e r t e d m a n y e a s y i l l u s t r a t i o n s o f i n t e g r a t i o n b yn u m e r i c a l w o r k s u c h as t h e e x e r c i s e s o n t h e B u l l E n g i n ea n d o n B e a m s a n d A r c h e s w h i c h a r e t o b e f o u n d i n m y b o o ko n A p p l i e d M e c h a n i c s . I c a n o n l y s a y t h a t I e n c o u r a g es t u d e n t s t o fin d i l l u s t r a t i o n s o f t h i s k i n d f o r t h e m s e l v e s ;a n d s u r e l y t h e r e m u s t b e s o m e l i m i t t o s p o o n f e e d i n g .

J O H N P E R R Y .

EOYAL COLLEOF, O F SCIENCE,LONDON,

16th Ma rch, 18117.


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T A B L E O F C O N T E N T S .

I NT R O D U CT O R Y R E M A R K S 1

C H A P . I . T H E S T U D Y O F .< :" ii

C H A P . I I . T H E C O M P O UN D I N T E R E S T L A W A N D T H E H A R M O N I CF U N C T I O N 1 6 1

C H A P . I I I . G E N E R A L D I F F E R E N T I A T I O N A ND I N T E G R A T I O N . 2 6 7


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CALCULUS FOR ENGINEERS.

I N T R O D U C T O R Y .

1 . T H E E N G I N E E R H A S U S U A L L Y N O T I M E FO R A G E N E R A L M A T H E M A T I C A L T R A I N I N G M O R E S T H E P I T Y A N D T H O S E Y O U N G E N G I N E E R SW H O H A V E H A D S U C H A T R A I N I N G D O NO T A L W A Y S F I ND T H E I R M A T H E M A T I C S H E L PF U L I N T H E I R P R O F E S S I O N . S U C H M E N W I LL , I H O P E ,F I N D T H I S B O O K U S E F U L , I F T H E Y C A N O NL Y G E T O V E R T H E N O T I O NT H AT B E C A U S E I T I S E L E M E N T A R Y , T H E Y K N O W AL R E A D Y ALL T H A T I TC A N T E A C H .

B U T I W R I T E M O R E PA R T I C U L A R L Y FO R R E A D E R S W H O H A V E H A DV E R Y LI TT LE M A T H E M A T I C A L T R A I N I N G A N D W H O A R E W I L L I N G T O W O R KV E R Y H A R D T O F I ND O U T H O W T H E C A L CU L US I S A P P L I E D I N E N G I N E E R I N G P R O B L E M S . I A S S U M E T H A T A G O O D E N G I N E E R N E E D S T O K N O WO N LY F U N D A M E N T A L P R I N C I P L E S , B U T T H A T H E N E E D S T O K N O W T H E S EV E R Y W E L L I N D E E D .

2 . M Y R E A D E R I S S U P P O S E D T O H A V E A N E L E M E N T A R Y K N O W L E D G E O F M E C H A N I C S , A N D I F H E M E A N S T O T A K E U P T H E E L EC T R I C A LP R O B L E M S H E I S S U P P O S E D T O H A V E A N E L E M E N T A R Y K N O W L E D G E O FE LE C T R I C A L M A T T E R S . A C O M M O N - S E N S E K N O W D E D G E O F T H E F E WF U N D A M E N T A L FA CT S I S W H A T I S R E Q U I R E D ; T H I S K N O W L E D G E I SS E L D O M A C Q U I R E D B Y M E R E R E A D I N G OR L I S T E N I N G T O L E C T U R E S ;O N E N E E D S TO M A K E S I M P L E E X P E R I M E N T S A N D TO W O R K E A S YN U M E R I C A L E X E R C I S E S .

I N M E C H A N I C S , I S H O U L D L I K E T O T H I N K T HA T T H E M E C H A N I C A LE N G I N E E R S W H O R E A D T H I S B O O K K N O W W H A T I S G I V E N I N T H EE L E M E N T A R Y P A R T S O F M Y B O O K S O N A P P L I E D M E C H A N I C S A N D T H ES T E A M A N D G A S E N G I N E . T H A T I S , I A S S U M E T HA T T H E Y K N O W

P 1


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2 C A L C U L U S F O R E N G I N E E R S .

T H E E L E M E N T A R Y F A C T S A B O U T B E N D I N G M O M E N T I N B E A M S , W O R KD O N E B Y F O R C ES A N D T H E E F F I C I E N C Y O F H E A T E N G I N E S . P O S S I B L YT H E B O O K M A Y C A U S E T H E M TO S E E K FOR S U C H K N O W L E D G E . I T A K EA L M O S T ALL M Y E X A M P L E S F R O M E N G I N E E R I N G , A N D A M A N W H OW O R K S T H E S E E A S Y E X A M P L E S W IL L FIND T H AT H E K N O W S M O S T O FW H A T I S C AL LE D T H E T H E O R Y O F E N G I N E E R I N G .

3 . I K N O W M E N W H O H A V E P A S S E D A D V A N C E D E X A M I N A T I O N SI N M A T H E M A T I C S W H O A R E V E R Y S H Y , I N P R A C T I C AL W O R K , O F T H EC O M M O N F O R M U L A E U S E D I N E N G I N E E R S ' P O C K E T - B O O K S . H O W E V E R G O O D A M A T H E M A T I C I A N A S T U D E N T T H I N K S H I M S E L F T O B E ,H E O U G H T T O P R A C T I S E W O R K I N G O U T N U M E R I C A L V A L U E S , T O FIN DFO R E X A M P L E T H E V A L U E O F ab B Y M E A N S O F A T A B L E O F L O G A R I T H M S ,W H E N a A N D b A R E A N Y N U M B E R S W H A T S O E V E R . T H U S T O FIN D^ ' 0 1 4 , TO FIN D 2 , 3 G " ) ~ - 1 ! , & C , T O T A K E A N Y F O R M U L A FR O M AP O C K E T - B O O K A N D U S E I T . H E M U S T N O T O N LY T H I N K H E K N O W S ;H E must R E AL LY D O T H E N U M E R I C A L W O R K . H E M U S T K N O W T H ATI F A D I S T A N C E 2 ' 4 5 4 H A S B E E N M E A S U R E D A N D I F O N E I S N O T S U R EA B O U T T H E LA ST FI GU RE, I T I S R A T H ER S T U P I D I N M U L T I P L Y I N G O RD I V I D I N G B Y T H I S N U M B E R T O G E T O U T A N A N S W E R W I T H M A N YS I G N I F I C A N T FI GU RES , O R T O S A Y T H AT T H E I N D I C A T E D P O W E R O F A NE N G I N E I S 3 2 4 ' 6 5 H O R S E P O W E R , W H E N T H E I ND I CA T O R M A Y B E I NE R R O R 5 P E R C E N T , OR M O R E . H E M U S T K N O W T H E Q U I C K W A Y O FF I N D I N G 3 2 1 G X 4 5 7 1 T O F OU R S I G N I F I C A N T FIG UR ES W I T H O U T U S I N GL O G A R I T H M S . H E O U G H T TO T E S T T H E A P P R O X I M A T E R U LE

( 1 +a)' l= 1 +n


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I N T R O D U C T O R Y . 3TRIANGLE AB G WHEN WE KNOW ONE SIDE AND ONE OF THE ACUTEANGLES. LEARN ALSO THAT THE SINE OF 1 3 0 IS POS ITIVE , AND THECOSINE OF 1 3 0 IS NEGAT IVE. ALS O TRY WITH THE BOOK OF TABLES IF

SIN ( A + B) = SIN A . COS B + COS A . SIN B,WHERE A AN D B ARE ANY TWO ANGLES YOU CHOOSE TO TAKE.TH ERE ARE THREE OTHER RULES LIKE TH IS. I N LIKE MANN ER THEFOUR WHICH WE OBTAIN B Y ADD ING T HESE FORMULAE AND SUBTRACTING T HE M, OF WHICH THIS IS ONE,

2 SIN A . COS j3 = SIN (A + /3) + SIN (A /3 );ALSO COS 2 A = 1 2 SIN 2 A = 2 cos- A I .

BEFORE READERS H AVE GONE FAR IN TH IS BOOK I HO PE THEY WILLBE INDUCED TO TAKE U P THE USEFUL (THAT IS , THE ELEMENTARYAND INTERESTING) PART OF TRIGONOMETRY, AND PROVE ALL RULES FORTHEMSELVES, IF THEY HAVEN'T DONE SO ALREADY.

CALCULATE AN ANGLE OF L'G DEGREES IN RADIANS ( 1 RADIAN ISEQUAL TO 5 7 - 2 9 6 DEGREES) ; SEE HOW MUCH THE SINE AND TANGENTOF THIS ANGLE DIFFER FROM THE ANGLE ITSELF. R E M E M B E R THATWHEN IN MATHEMATICS WE SAY SIN x, x IS SUPPO SED TO BE INRADIANS.

I DO NOT EXPECT A MAN TO KNOW M UCH ABOUT ADVANCEDALGEBRA, BUT H E IS SU PPO S ED TO B E ABLE TO GIV E THE FACTORS OFa? + 7 # + 1 2 OR OF


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4> C A L C U L U S F O R E N G I N E E R S .E L E M E N T A R Y C A LC U LU S W O R K I S I N I N D U C I N G S T U D E N T S T O A G A I NU N D E R G O T H I S D R I L L I N G .

B U T T H E E N G I N E E R N E E D S N O A R T IF I CI AL M E N T A L G Y M N A S T I C SS U C H A S I S F U R N I S H E D B Y G E O M E T R I C A L C O N I E S , O R T H E U S U A LE X A M I N A T I O N - P A P E R P U Z Z L E S , OR B Y E V A S I O N S O F T H E C A L C U L U ST H R O U G H I N F I N I T E W O R R Y W I T H E L E M E N T A R Y M A T H E M A T I C S . T H ER E S U L T O F A F AL S E S Y S T E M O F T R A I N I N G I S S E E N I N T H I S , T H A T N O TO N E G O O D E N G I N E E R I N A H U N D R E D B E L I E V E S I N W H A T I S U S U A L L YC A L L E D T H E O R Y .

4 . I A S S U M E T H A T E V E R Y O N E O F M Y R E A D E R S I S T H O R O U G H LYW E LL A C Q U A I N T E D A L R E A D Y W I T H T H E F U N D A M E N T A L N O T I O N O F T H EC A L C U L U S , O N LY H E D O E S N ' T K N O W I T I N T H E A L G E B R A I C F O R M . H EH A S A P E I ' F E C T K N O W L E D G E O F a rate, B U T H E H A S N E V E R B E E NA C C U S T O M E D TO W R I T E ~ ; H E H A S A P E R F E C T K N O W L E D G E O F A NA R E A , B U T H E H A S N O T Y E T L EA RN T T H E S Y M B O L U S E D B Y U S ,jf (a" ).da\ H E H A S T H E I D E A , B U T H E D O E S N O T E X P R E S S H I SI D E A I N T H I S F O R M .

1 A S S U M E T H A T S O M E O F M Y R E A D E R S H A V E P A S S E D D I FF I CU LTE X A M I N A T I O N S I N T H E C A L C U L U S , T H AT T H E Y CA N D I F F E R E N T I A T E A N YF U N C T I O N O F x A N D I N T E G R A T E M A N Y ; T H A T T H E Y K N O W H O W T OW O R K ALL S O RT S O F D I FFI C UL T E X E R C I S E S A B O U T P E D A L C U R V E S A N DR O U L E T T E S A N D E L L I P T I C I N T E G R A L S , A N D TO T H E M A LS O I H O P E TOB E O F U S E . T H E I R D I FF IC U LT Y I S T H I S , T H E I R M A T H E M A T I C A L K N O W L E D G E S E E M S T O B E O F NO U S E TO T H E M I N PR A CT I CA L E N G I N E E R I N GP R O B L E M S . G I V E T O T H E I R a' a A N D Y ' S A P H Y S I C A L M E A N I N G ,O R U S E P ' S A N D v' s I N S T E A D , A N D W H A T W A S T H E E A S I E S T B O O KE X E R C I S E B E C O M E S A D IF F I C UL T P R O B L E M . I K N O W S U C H M E NW H O H U R R I E D L Y S K I P I N R E A D I N G A B O O K W H E N T H E Y S E E A

O R A S I G N O F I N T E G R A T I O N .

5 . W H E N I S T A R T E D TO W R I T E T H I S B O O K I T H O U G H T TO P U TT H E S U B J E C T B E F O R E M Y R E A D E R S A S I H A V E B E E N A B L E , I T H I N K .I H A V E B E E N T O L D V E R Y S U C C E S S F U LL Y , T O B R I N G I T B E F O R E S O M EC L A S S ES O F E V E N I N G ' S T U D E N T S ; B U T M U C H M A Y B E D O N E I NL E C T U R E S W H I C H O N E I S U N A B L E T O D O I N A C O L D - B L O O D E D F A S H I O NS I T T I N G A T A T A B L E . O N E M I S S E S T H E I N T E L L IG E N T E Y E S O F A NA U D I E N C E , W A R N I N G O N E T H A T A LI TT LE M O R E E X P L A N A T I O N I S N E E D E D


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I N T R O D U C T O R Y , 5

O R T HA T A N I M P O R T A N T I D E A H A S A L R E A D Y B E E N G R A S P E D . A NI D E A C O UL D B E G I V E N I N T H E M E R E D R A W I N G O F A C U R V E A N DI LL US T R A TI O NS C H O S E N F R O M O B J E C T S A R O U N D T H E L E C T U R E - R O O M .

L E T T H E R E A D E R S K I P J U D I C I O U S L Y ; L E T H I M W O R K U P NOP R O B L E M H E R E I N W H I C H H E H A S N O P R O F E S S I O N A L I N T E R E S T . T H EP R O B L E M S A R E M A N Y , A N D T H E B E S T T R A I N I N G C O M E S F R O M T H EC AR E FU L S T U D Y O F O N L Y A F E W O F T H E M .

T H E R E A D E R I S E X P E C T E D T O T U R N B A C K O F T EN T O R E A D A G A I NT H E E A R L Y P A R T S .

T H E B O O K W O U LD B E U N W I E L D Y I F I I N C L U D E D A N Y B U T T H EM O R E I N T E R E S T I N G A N D I L L U S T R A T I V E O F E N G I N E E R I N G P R O B L E M S . IP U T O FF FO R A F U T U R E O C C AS I O N W H A T W O U L D P E R H A P S T O M A N YS T U D E N T S B E A M O R E I N T E R E S T I N G P A R T O F M Y S U B J E C T , N A M E L Y ,I LL U S T R AT IO NS F R O M E N G I N E E R I N G ( S O M E T I M E S C A L L E D A P P L I E DP H Y S I C S ) O F T H E S O L U T IO N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S .M A N Y P E O P L E T H I N K T H E S U B J E C T O N E W H I C H C AN NO T B E T A U G H TI N A N E L E M E N T A R Y F AS H I O N , B U T L O R D K E L V I N S H O W E D M E LONG -A GO T H AT T H E R E I S NO U S E F U L M A T H E M A T I C A L W E A P O N W H I C H A NE N G I N E E R M A Y N O T L EA R N T O U S E . A M A N LE AR N S T O U S E T H EC A L C U L U S A S H E L E A R N S T O U S E T H E C H I S E L O R T H E F I L E O N A C T U A LC O N C R E T E B I T S O F W O R K , A N D I T I S O N T H I S I D E A T H A T I A CT I NT E A C H I N G T H E U S E O F T H E C A L C U L U S T O E N G I N E E R S .

T H I S B O O K I S N OT M E A N T TO S U P E R S E D E T H E M O R E O R T H O D O XT R E A T I S E S , I T I S R A T H E R A N I N T R O D U C T I O N T O T H E M . I N T H EFIR ST C H A P T E R O F 1 G 0 P A G E S , I D O NO T A T T E M P T T O D I F F E R E N T I A T EO R I N T E G R A T E A N Y F U N C T I O N O F :t% E X C E P T X " . I U T H E S E C O N DC H A P T E R I D E AL W I T H e" x, A N D S I N (u .v + c). T H E T H I R D C H A P T E RI S M O R E D I F FI C U LT .

F O R T H E S A K E O F T H E T R A I N I N G I N E L E M E N T A R Y A L G E B R A I CW O R K , A S M U C H A S FO R U S E I N E N G I N E E R I N G P R O B L E M S , I H A V EI N C L U D E D A S E T O F E X E R C I S E S O N G E N E R A L D I F F E R E N T I A T I O N A N DI N T E G R A T I O N .

P A R T S I N S M A L L E R T Y P E , A N D T H E N O T E S , M A Y B E F O U N D T OOD IFFI CU LT B Y S O M E S T U D E N T S I N A FI RS T R E A D I N G O F T H E B O O K . A NO C CA S IO N AL E X E R C I S E M A Y N E E D A L IT T LE M O R E K N O W L E D G E T H A NT H E S T U D E N T A L R E A D Y P O S S E S S E S . H I S R E M E D Y I S T O S K I P .


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C H A P T E R T.x n .

6 . E V E R Y B O D Y has a l ready the not ions of C o -o rd i n at eG e o m e t r y a nd u se s s q u a r e d p a p e r . S q u a re d p ap er m ayb e b ought at sevenpencc a q ui re: people who arc ignorantof this fact and who pay sevenpence or fourteen pence ashee t for it mus t have too gre at an idea of i ts value to useit properly.W hen a m e r c h a n t has in his off ice a sheet of squaredpaper with points lying in a curve which he adds to day byday, each point showing the price of iron, or copper, or cottonyarn or silk, at any date, h e is usin g Co-ordin ate G eom etry.Now to w hat uses does he put such a curve ? 1. A t an ydate he sees wh at th e price was. 2. H e sees by th e slope ofhis curve the rate of increase or fall of the price. 3 . I f h eplots other things on the same sheet of paper at the samedates he will note what effect their rise and fall have uponthe price of his m aterial , and this may enable him to prophesy and so mak e money. 4 . Exam in ation of his curve forthe past will enable him to prophesy with more certaintythan a man can do who has no records.O b serve t h at a n y p o i n t r e p r e s e n t s t w o t h i n g s ; i tshorizontal distance from some standard line or axis is calledone co-ordinate, we generally call it the .r co-ordinate and itis measured horizontally to the right of the axis of y ; somepeople call it the abscissa; this represents tim e in his case.The other co-ordinate (we usually call it the y co-ordinate orthe ordinate, simply), the vertical distance of the point abovesome standard line or axi s; this represents his price. In th enewspaper you will f ind curves showing how the therm om eterand b arom eter are rising and falling. I once read a cleverarticle upon th e way in which the En glish population andwealth and taxes were increasing; the reasoning was very


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S Q U A R E D P A P E R .

difficult to follow. On takin g th e author's figures howeverand plotting them on squared paper, every result which hehad laboured so much to bring out was plain upon thecurves, so th at a boy could und erstand th em . Possib ly thi sis the reason why some writers do not publish curves: if theydid, there would be little need for writing.7 . A man making e xp e ri m e n ts is usual ly f inding outhow one thing which I shall call y depends upon some otherthing which I shall call x. Thus the pressure p of saturatedsteam (water and steam present in a vessel but no air or

other f luid) is always the same for the same temperature.A curve drawn on squared paper enables us for any giventemperature to find the pressure or vice versa, but it showsthe rate at which one increases relatively to the increase ofthe other and much else. I do not say th at the curve is alwaysbetter than the table of values for giving information ; someinformation is better given by the curve, some by the table.Observe that when we represent any quantity by the lengthof a line wo represen t it to some scale or other : 1 inchrepresents 10 lbs. per square inch or 20 degrees centigradeor something- else; it is always to scale and according to aconvention of some kind, for of course a distance 1 inch is avery different thing from 20 degrees centigrade.When one has two columns of observed numbers to ploton squared paper one docs it , 1 . T o see if th e points l ie inany regular curve. I f so, the simp ler the curve th e simpleris the law that we are likely to find . 2. T o correct errors ofobservation. Fo r if the points lie ne arly in a simple regul arcurve, if we draw the curve that lies most evenly among thepoints, using thin battens of wood, say, then it may be takenas probable that if there were no errors of observation thepoints would lie exa ctly in such a curve. N ote tha t whena point is 5 feet to the right of a line, we mean that it is5 feet to the left of the line. I h ave learnt by long experience that it is worth while to spend a good deal of timesubtracting from and multiplying one's quantities to f it thenumbers of squares (so that the whole of a sheet of paper isneeded for the points) before beginning to plot.

Now let the reader buy some squared paper and withoutasking help from anyone let him plot the results of some


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8 C A L C U L U S F O R E N G I N E E R S .observations. Le t him take for exampl e a W hi tak er'sAl man ack and plot from it some sets of numb ers ; the averagetemp erature of every month last ye ar ; the National D eb tsince 1688; the present value of a lease at 4 per cent, forany number of years ; the capita l invested in Rai lways since1849 ; anything will do, but he had better take things inwhich he is interested. I f he has made lab oratory ob servations he will have an absorbing interest in seeing what sortof law the squared paper gives him.

8. As the observations may be on pressure p and temperature t, or p a n d volume v, or v an d t, or Indicated HorsePower and Useful Horse Power of a steam or gas engine, oramperes and volts in clectricit j - , and we want to talk gene rallyof any such pair of quantities, I shall use .-


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' G R A P H ' E X E R C I S E S . 9

take x 2, then y = 2 + 1 3 3 3 = 2133; an d so on. N o w plotthese values of x and y on your sheet of squared paper. T h ecurve is a parabola.

I I . Draw the curve y = 2 Ix + J-^x" which is also aparabola, in the same wa y, on the same sheet of paper.

I I I . Draw the curve a-tj = 120. Now i f x = 1, y = 120 ;if x = 2, y = 60 ; if x = 3, y = 40 ; if x = 4, y = 30 an d so o n :this curve is a rectangular hyperbola.

IV. Draw yx1-414 = 100 or y = 1 0 0 A -- 1 - 4 1 4 . If the studentcannot calculate y for any value of x, lie does not know howto use logarithms and the sooner he does know how to uselogarithms the better.

V. Draw y = ax' 1 where a is an}" convenient number. Iadvise th e student to spend a lot of time in drawing membersof this great family of useful curves. Le t hi m try = 1(he drew this in III . above), n= 2, = 1A-, = | , = 0 1 ,n 0, n = \, n f, n = 1, n = \\, u='2 (this is No. I. above),n = 3, n = 4 &c.

V I . Draw y = a s i n ( b x + c) taking any convenientnumbers for a, b an d c.

Advice. A s bx + c is in radians (one radian is 57-2958degrees) and the books of tables usua lly give an gles indegrees, choose numbers for 6 and c which will make th earithmet ical wor k easy. Th us ta ke b= 1 -~ 114-6, take c th enumber of radians which correspond to say 30

^this is ~ or -.5236j .Let a = 5 say . N o w le t x = 0,10, 20, & c , and calculate y.

/ 6 \Thus when # = 6 , y = 5 sin L ^ .q + '5236 ); but if theangle is converted into degrees we have

y = 5 sin (6 + 30 degrees) = 5 sin 33 = 2-723.Having drawn th e above curve, notice wha t cha nge would

occur if c were changed to 0 or ^ or ^ or ~ . Again, i f awere changed. More tha n a week ma y be spent on this curve,very profitably.


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10 C A L C U L U S F O R E N G I N E E R S .V I I . Draw y = a e b x . T ry b = l and a = 1 ; try othervalues of a an d b; take at least two cases of n egati ve valuesfor b.In the above work, get as l itt le help from teachers aspossible, but help from fellow students will be very usefulespecially if it leads to wrangling about the subject.The reason why I have dwelt upon the above seven casesis thi s: S tud en ts learn usually to differentiate and in tegratethe most complicated expressions: but when the very simplestof these expressions comes before them in a real engineering

problem the y fight shy of it. Now it is very seldom that anengineer ever has to face a problem, even in the most intricate part of his theoretical work, which involves a knowledgeof more fun ctions than these threey = ax n, y = a sin (bx + c), y = ae' ' *,

but these three must be thoroughly well understood and theengineering student must look upon the study of them as hismost important theoretical work.Attending to the above three kinds of expression is astude nt's real business. I see no reason, however, for his nothaving a little amusement also, so he may draw the curves

+ if = 2.5 (C ircl e), |^ + = 1 (Ell ip se ),^5 ~ 16 = 1 ( H j ' P e r b o l a ) '

and some others mentioned in Ch apter I I I . , but f rom theengineer's point of view these curves are comparatively uninteresting.10. Having studied y = e" nx an d y = b sin (cx + g) astudent will find that he can now easily understand one ofthe most important curves in engineering, viz:

y = b e - a s E s in ( cx + g ) .He ought first to take such a curve as has already beenstudied by him, y b sin (cx + g) ; plot on the same sheet ofpaper y = e" ax ; and multiply together the ordinates of thetwo curves at many values of x to f ind the ordinate of the


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E Q U A T I O N S T O L O C I . 11new curve. The curve is evidently wavy, y reaching maximumand minimum values; y represents the displacement ofa pendulum bob or pointer of some measuring instrumentwhose motion is damped by fluid or other such friction, so beingthe time, and a student will understand the curve much betterif he makes observations of such a motion, for example witha disc of lead immersed in oil vibrating so slowly underthe action of torsional forces in a wire that many observations of its angular position (using pointer and scaleof degrees) which is called y, x being the time, may bemade in one swing. The distance or angle from an extremeposition on one side of the zero to the next extreme positionon the other side is called the length of one swing. TheNapierian logarithm of the ratio of the length of one swing tothe next or one tenth of the logarithm of the ratio of thefirst swing to the eleventh is evidently a multiplied by halfthe periodic time, or it is a multiplied by the time occupiedin one siviny. This logarith mic de cre me nt as it is called,is rather important in some kinds of measurement.

11. When by moans of a drawing or a model wo are able to find thepath Of any point and where it is in its path when we know theposition of some other point, wo arc always able to get the sameinformation algebraically.

Example (1). A point F and a straight line 7)1) being given ; whatis the patli of a point P when it moves no that its distance from thepoint F is always in the same ratio to its distance from the straight line ?Thus in the figure let 1'F^e x I 'D.. . ( 1 ) , where e is a constant.Draw EFX at right angles to DO.If the distance I'D is called x andthe perpendicular VG is y; ourproblem is this ; What is the equation connecting x and y ? Xow allwe have to do is to express ( 1 ) interms of x and // Let EF be called a.Thus

pf= \)p& 1+F


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1 2 C A L C U L U S F O R E N G I N E E R S .Example (2 ). TH E CIRCLE AP Q ROLLS ON THE STRAIGHT LINE O XWH AT IS THE PATH OF ANY POINT P ON THE CIRCUMFERENCE ? TF WHEN P

TOUCHED THE LINE IT WAS AT 0 , LET O A ' AM I OY BE THE AXES, AND LET SPBE x AND PT HE y. LOT THE RADIUS OF THE CIRCLE BE a. LET THE ANGLEPC'Q BE CALLED . ' DRAW CB , PERPENDICULAR TO PT. OBSERVE THATPit=a . SIN PCB = a SIN ((/; -


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M E C H A N I S M S . 1 3IT IS WELL TO REMEMBER IN ALL SUCH PROBLEMS THAT IF WE PROJECT ALLTHE AIDES OF A CLOSED FIGUREUPON ANY TWO STRAIGHT LINES, WE GET TWO INDEPENDENT EQUATIONS. PROJECTING ON THE HORIZONTAL WE SEE THATs + lcox


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14 C A L C U L U S F O R E N G I N E E R S .position. The complete path of 1" would be afigureof 8. 3. 1'indthe equation to the path of a point in the middle of an ordinary connecting rod. 4. A, the end of a link, moves in a straight path COC,0 being the middle of the path, with a simple harmonic motionOA=anmpt, where t is time; the other end B moves in a straightpath OBD which is in a direction at right angles to COC; what is B' smotion 1 Show that it is approximately a simple harmonic motion oftwice the frequency of A. 5. In any slide valve gear, in which thereare several links, &c. driven from a uniformly rotating crank ; notethis fact, that the motion of any point of any link in any particulardirection consists of a fundamental simple harmonic motion of thesame frequency as the crank, together with an octave. The properstudy of L i n k M o t i o n s a n d R a d i a l v a l v e g e a r s from th ispoint of view is worth months of one's life, for this contains the secretof why one valve motion gives a better diagram than another.Consider for example the Hackworth gear with a curved and with astraight slot. What is the difference ? Sec Art. 122.

1 2. P l o t t e d p o i n t s l y i n g i n a s t r a i g h t l i n e . P ro of swill come later; at f irst the student ought to get well acquainted with the thin g to be proved. I have known boysable to prove mathematical propositions who did not reallyknow what they had proved till years afterwards.Take any expression like y = a + bx, where a an d b arenumb ers. Th us let y = 2 + l^x. Now take x = 0, x = l,x = 2, x = 3, & c. and in each case calculate the correspondingvalue of y. Plot the corresponding values of x an d y as theco-ordinates of points on squared paper. Yo u will f ind tha tthey lie exactly in a straight l ine. Now tak e say y = 2 + 3xor 2 + h,x or 2 ^x or 2 ' ix and you will find in every casea straight l ine. Men who think they know a litt le abou tthis subject already will not care to take the trouble and ifyou do not find yourselves interested, I advise you not totake the trouble either; yet I know that it is worth yourwhile to take the trouble. J ust notice th at in every case Ihave given you the same value of a and consequently allyour lines have some one thin g in common. W h at is it ?Take this hint , a is the value of y when x = 0.Again, try y = 2 + l y = 1 + 1 \x, y = 0 + H x ,

y = - l + l% x, y = - 2 + l i . r ,and so see what it means when b is the same in every case.You will f ind that all the lines with the same b have the


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S L O P E O F A LINE. 1 5same slope and indeed I am in the habit of calling b th eslope of the line.I f y = a + bx, when x = x- i, find y and call it yuwhen x = , i \ + 1, find y and call it y.I t is easy to show that y. 2 yl = 6. So tha t what I mean byth e slope of a straight line is its rise for a horizontal distance1. (N ote th at when w e say th at a road rises ^ or 1 in 20,we mean 1 foot rise for 20 feet along th e sloping road. Th usfo is the sine of the angle of inclination of the road to thehorizontal; whereas our slope is measured in a different way).Our slope is evidently the tang ent of the inclination of thel ine to the horizontal. Look ing upon y as a quantity whosevalue depends upon that of x, observe that the rate of increase of y relatively to the increase of x is constant, beingindeed b, the slope of the line. T he symb ol used for thi srate is . Observe tha t it is one sym bol; it does not meanax-- X ^ . T ry to reco llect the s tat em en t th at i f ?/= a+ b.c,d x x J J >

= b, and that i f = b, then it follows that y = A + bx,ax ax Jwhere A is some constant or other.Any equation of the first degree connecting x and ysvich as Ax + By=G where A, B and G are constants, canG Abe put into the shape y = = - x, so that it is the equationAto a straig ht line whose slope is and which passesGthrough the point whose x = 0, whose y = T , called point(0 , -gj T h u s


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16 CALCULUS FOR ENGINEERS.

XFig. 5.

b tan BOX. I f BE is any line at right angles to the first,its slope is tan BEX or tan BEG or cot BCE or ~ .

So that y = A ^ x is typical of all lines at right anglesto y = a + bx; A being any constant.

3 . W here do the two straight l ines Ax + Bi/+ O0and Mac + Ny + S = 0 m ee t? Answer, In the point whose mand y satisfy both the equations. W e have therefore to dowhat is done in Ele m entary Algebra, solve simultaneousequations.4 . W he n tan o and tan/3 are known, it is easy to findtan ( a and hence when th e straig ht lines y=a + bxand y = m + n.v are given, it is easy to find the angle betweenthem.5. The line y = a + bx passes through the points x~\,y = 2, and x = 3 , y=\, find a and b.

told that it passes through the point whose x = 3, and whosey = 2, what is the equation to the line ? Let the slope be0-35.The equation is y = a. + 0'35.'i'. where a is not known.B u t (3 , 2) is a point on the line, so tha t 2 = a + 0 '3o x 3 ,or a = 0"95 and hence the line is y = 0'95 + 0 35a\2. What is the slope of any line at right angles toyz=a + bw? L e t AB be the given line, cutting OX in C. Th en


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E M P I R I C A L F O R M U L A E . 176. A line y = a + bx is at right angles to y = 2 + &e and

passes through the point * = 1 , y = 1. Fin d a and 6.14 . O b t ai n i n g E m p i r i c a l F o r m u l a e .When in the laboratory we have made measurements of twoquantities which depend upon one another, we have a table showingcorresponding values of the two, and we wish to see if there is a simplerelation between them, we plot the values to convenient scales as theco-ordinates of points on squared paper. If some regular curve (acurve without singular points as I shall afterwards call it) seems as ifit might pass through all the points, save for possible errors ofmeasurement, we try to obtain a formula y = f(x), which we may callthe law or rule connecting the quantities called y and x.If the points appear as if they might lie on a straight line, astretched thread may be used to help in finding its most probableposition. There is a tedious algebraic method of finding the straightline which represents the positions of the points with least error, butfor most engineering purposes the stretched string method is sufficiently accurate.If the curve seems to follow such a law as y = a + bx 2, plot y andthe square of the observed measurement, which we call x, as the co-ordinates of points, and see if they lie on a straight line. If the curve

seems to follow such a law as y = f^ T fJ ^ (1)' which is the same asy. + by = a, divide each of the quantities which you call y by the corresponding quantity x ; call the ratio A*. Now plot the values of A' andof y on squared paper; if a straight line passes throxigh the plottedpoints, then we have such a law as X =A + By, or - = A+By, orV = T K ~ > so that (1) is true.3 l-Bx' wUsually we can apply the stretched thread method to find theprobability of truth of any law containing only two constants.Thus, suppose measurements to be taken from the expansion partof a gas engine indicator diagram. I t is important for many purposesto obtain an empirical formula connecting p and v, the pressure andvolume. I always find that the following rule holds with a f air amountof accuracy p v s = C where s and C are two constants. We do notmuch care to know C, but if there is such a rule, the value of s is veryimportant*. To test if this rule holds, plot log^ and log v as the co-

* There is no known physical reason for expecting such a rule to hold.AtfirstI thought that perhaps most curves drawn at random approximatelylike hyperbolas would approximately submit to such a law as yx n=C, but Ifound that this was by no means the case. The following fact is worthmentioning. When my studentsfind, n carrying out the above rule thatlogp and log i' do not lie in a straight line, I find that they have1'. 9


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18 C A L C U L U S F O R E N G I N E E R S .ordinates of points on squared paper (common logarithms will do).If they lie approximately in a straight line, we see thatlog p+s log v -ca constant, and therefore the rule holds.When we wish to test with a formula containing three independentconstants we can often reduce it to such a shape as

Av+Bic+Cz=\ (2),i i nM , , , -v a + bxwhere -c, v\ g contain * and y m some shape. 1 hus to test rr y=- ,

we have y + cxy = a + bx, ov + -.n/ x\. Here y itself is the old v.J a it a xi/is the old M I , and x itself is the old z.If (2) holds, and if


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SLOPEOFACURVE 19theyare askedtofindthe rue (3) whchmost accuratelyrepresentsp&nd 8 between sayp=7 l b . per sq inch andp=150. He whogetsastraight line lyngmost eveny(judgngbythe eye) amongthe points,when\ogpandlog(#+/3) are usedas co-ordnates, has usedthe bestvalue of 0. The methodmayhe refineduponbyingenous students.(See endof Chap I.)15. We have nowtoremember that if y a+bx, then- ^ = 6, andif ~-=5 then u = A+bx, where Ais someax axconstant.

Let us prove this algebraicallyIfy=a+bx Take aparticular value ofxandcalculate

y Nowtake a newvalue of x call itx+Sx and calculatethe newy call it y+By

y + By = a + b (x -f Sx).Subtract y=a+ bxandwe get

8y =bSx, or ~- = b,and however small Sxor Symaybecome their ratioisb we

Fig G.16. In the curve of fig 6there is p o s i t i v e s l o p e

(jf increases asx increases) inthe parts AB, DF andHI and22


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20 C A L C U L U S F O R E N G I N E E R S .n e g a t i v e s l o p e (y diminishes as x increases) in the partsBD and FH. The , s lope i s 0 a t B and F which are calledpoints o f maximum or po in ts where y is a maximum; and i tis also 0 at D an d H which are po in ts o f m in im um . Thepoint E is one in which the slope ceases to increase andbegins to diminish : i t is a point of inf lexion.

No tice that if we want to know the slope at the point Pwe first choose a point F whichis near to F. ( Imag ine tha tin fig. 6 the little portion ofthe curve at P is magnified athousand tim es.) Call P8~x,PQ=y;NF=z+Bx,FL=y+By,so tha t PM = Bx, FM= By. NowFM jP M ov Sy/Sx is the averageslope between P and F. I t istan FPM. Imagine the samesort of figure drawn but for apoint F' nearer to P. Again,another, sti l l nearer P. O b serve that the straight l ine FP or F' P or F" P gets graduallymore and more nearly what we mean by the tangent to thecurve at P. In every case By/Bx is the tangent of the anglewhich the line FP or F' P or F' P makes with the horizontal,and so we see that in the limit the slope of the line or dyjdxat P is the tangent of the angle which the tangent at Pmakes with the axis of X . I f then, instead of judgingroughly by th e eye as we did j u st now in di scussing fig. 6, wewish to measure very accurately the slope at the point P ; Note that the slope is independent of where the axis of X is,so long as it is a horizontal line, and I take care in using myrule here given, to draw OX below the part of the curvewhere I am studying the slope. D raw a tan gen t PR tothe curve, cutt ing OX in R. Then the slope is tan PRX.I f drawn anil lette red according to my ins tructions, obse rvet h a t PRX is always an acute angle when the slope ispositive and is always an obtuse augle when the slope isnegative.

Do not forget that the slope of the curve at any pointmeans the rate of increase of y there with regard to x, an d


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W H A T I S S P E E D ? 21THAT WE M A Y CALL IT THE slope OF THE CURVE OR TAN PRQ OR BYd vTHE SYMBOL ^j - OR " TH E DIFFERENTIAL COEFFICIENT OF Y W IT HREGARD TO X , " AND ALL THESE ME AN THE SAM E THING.

EVERY ONE KNOWS WHAT IS MEANT WHEN ON GOING UP AHILL ONE SAYS THAT THE SL OP E IS CHANGING, THE SLOPE IS D IM IN IS H ING, THE SLOPE IS INCREASING ; AND IN TH IS KNOWLEDGE HE ALREADYPOSSES SES TH E FUNDAMENTAL IDEA OF THE CALCULUS.

1 7 . W E ALL KNOW WHAT IS MEANT WH EN IN A RAILWAY TRAIN WESAY " W E A RE G O I N G AT 3 0 M I L E S P E R H O U R . " D O W E M E ANTHAT WE HAVE GONE 3 0 MILE S IN THE LAST HOUR OR THAT WE AREREALLY GOING 3 0 MILES IN THE NEXT HOUR ? CERTAINLY NOT. W EMAY HAVE ONLY LEFT THE TERMINU S 1 0 MIN UT ES AGO ; THERE MA Y BEAN ACCIDENT IN THE NEXT SECOND. W NAT WE ME AN IS MERELY THIS,THAT THE LAST DISTAN CE OF 3 MILE S WAS TRAVERSED IN THE TENTHOF AN HOUR, OR RATHER, THE LAST DISTANCE OF 0 O 0 0 3 MILE S WASTRAVERSED IN O'OOOOL HOUR. T H IS IS NOT EXACTLY RIGH T; IT ISNOT TILL WE TAKE STILL SHORTER AND SHORTER DISTANCES AND DIVIDEB Y THE TIM ES OCCUPIED THAT WE APPROACH THE TRUE VALUE OFTHE SPE ED . T H U S IT IS KNOW N THAT A BO DY FALLS FREELYVERTICALLY THROUGH THE FOLLOWING DISTANCES IN THE FOLLOWINGINTERVALS OF T IM E AFTER TWO SECON DS FROM REST, AT LO ND ON .THAT IS BETWEEN 2 SECONDS FROM REST AND 2 1 OR 2 '0 1 OR 2 '0 0 1 ,THE DISTANCES FALLEN THROUGH ARE GIVE N. E AC H OF THESEDIVIDED B Y THE INTERVAL OF TIM E GIV ES THE AVERAGE VELOCITYDURING THE INTERVAL.INTERVALS OF T IM E IN SECONDS -1DISTAN CES IN FEET FALLEN THROUGH 6 ' 6 0 1

0 1 - 0 0 1 6 4 5 6 J - 0 6 4 4 1 6

AVERAGE VELOCITIES 1 6 6 0 1 ' 6 4 ' 5 6 : 6 4 ' 4 1 6W E SEE THAT AS THE INTERVAL OF T IM E AFTER 2 SECONDS IS

TAKEN LESS AND LESS, THE AVERAGE VELOCITY DURING THE INTERVALAPPROACHES MORE AND MORE THE TRUE VALUE OF THE VELOCITY AT2 SECONDS FROM REST WHICH IS EXACTLY 6 4 ' 4 FEET PER SECOND.

W E M AY FIND THE TRUE VELOCITY AT AN Y TIM E WHEN WE KNOWTHE LAW CONNECTING s AN D t AS FOLLOWS.

L E T 8 = 1 6 " I T 2 , TH E WELL KNO WN LAW FOR BO DI ES FALLINGFREELY AT LO ND ON . I F t IS G IVEN OF AN Y VALUE WE CAN CALCULATE


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s. I f t has a slightly greater value called t + St (here Stis a symbol for a small jxxrtion of time, it is not S x t, but avery different thing), and if we call the calculated spaces+Ss, then s + Ss = 1 6 1 (t + Sty o r 1 6 1 [t 1 + 2t. St + (Sty}.Hence, subtract ing, Ss = 1 6 1 [2t. ct + (St)-}, and this formulawill enable us to calculate accurately the space Ss passedthrough between the time t and the time t + St. The averagevelocity during thi s interval of t ime is Ss -=- St or^ = 3 2 " 2 + l ( J l S f .otPlease n otice th at thi s is absolutely co rre ct; there is novagueness about it .

Now I come to the important idea; as St gets smallerSsand smaller, ^- app roaches more and more ne arly 32 -2, theotother term JQ-lSt becoming smaller and smaller, and hencewe say tha t in th e l im i t , SsiSt is truly 32-2L The l imit ingSs ' " ds 1value of ^ as St gets small er and smaller is called , or theSt atrate of change of s as t increase s, or the differential coefficientof s with regard to t, or it is called the velocity at the time f.Now surely there is no such grea t diff iculty in catchingthe idea of a l imi ting value. S ome people have the notionthat wo are stating something that is only approximatelyt r u e ; it is often because their teacher will say such things as" re j ec t 161 St because it is small ," or "let dt be an infinitelysmall amount of t i m e " and they proceed to divide somethingby it , showing that although they m ay reach the age of

M ethuselah they will nev er have the common sense of anengineer.An other trouble is introduced by people saying " letS * dsSt 0 and ^- or is so and so." T he tru e statem en t is, " asSt at gSt gets smaller and smaller without limit, ^- approaches more

and more nearly the finite value 32 2'," and as I have alreadysaid, everybody uses the important idea of a limit every dayof his life,


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S L O P E A N ] ) S P E E D . 2 3

From the law connecting s and t, if we find ^ or thevelocity, we are said to d i f f e r e n t i a t e s with regard to thet ime t. When we are g iven dsjdt and we reverse the aboveprocess we are said to i n t e g r a t e .If I were lecturing I might dwell longer upon the correctness of the notion of a rate th at one already has, and bymaking man y sketche s il lustrate my meaning. B u t one ma ylisten in ten tly to a lectu re which seems dull enough in abook. I will, therefore , ma ke a virtu e of nece ssity and say

tha t my readers can il lustra te my meaning perfectly well tothem selves if they do a l itt le th inkin g about it . Afte r allmy great aim is to ma ke them less afraid than they used to beof such symbols as dyjdx an d fy . dx.18 . Given and t in any kind of motion, as a set of numbers. How do we study the motion ? Fo r exam ple, ima ginea Bradshaw 's Railw ay Guide which not merely gave a fewstations, but some hundred places between Euston and Rugby.Th e entrie s m igh t be like t his : * would be in miles, t in

hours and minutes. .? = 0 would mean Euston.t

0 10 o'clock3 1 0 . . 105 1 0 . . 157 1 0 . . 2 07 1 0 . . 2 39 1 0 . . 2812 TO..33& c.

One m ethod is th is : plot t (take times after 10 o 'clock)horizontally and s vertically on a sheet of squared paper anddraw a curve through the points.The slope of this curve at any place represents the velocityof the train to some scale which depends upon the scales fors an d t.


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2 4 C A L C U L U S F O R E N G I N E E R S .

OBSERVE PLACES WHERE THE VELOCITY IS GREAT OR SMALL.B E T W E E N t = 1 0 . 2 0 AND = 1 0 . 2 3 OBSERVE THAT THE VELOCITYIS 0. I N D E E D THE TRAIN HAS PROBABLY STOPPED ALTOGETHER.T O B E ABSOLUTELY CERTAIN, IT WOULD BE NECESSARY TO GIVE SFOR EVERY VALUE OF t, AND NOT MERELY FOR A FEW VALUES. ACURVE ALONE CAN SHOW EVERY VALUE. I DO NOT SAY THAT THETABLE M A Y NOT BE MORE VALUABLE THAN THE CURVE FOR A GREATM A N Y P U R P O S E S .

I F THE TRAIN STOPPED AT ANY PLACE AND TRAVELLED TOWARDSEUS TO N A GAIN , WE SHOULD HAVE NEGATIVE SLOPE TO OUR CURVEAND NEGATIVE VELOCITY.

NOTE THAT acce l er a t ion BE IN G RATE OF CHANG E OF VELOCITYW I T H T I M E , IS INDICATED B Y THE RATE OF CHANGE OF THE SLOPE OFTHE CURVE. W H Y NOT ON THE SAM E SHEET OF PAPE R DRAW A CURVEWHICH SHOWS AT EVERY INSTANT THE v e l o c i t y OF THE TRAIN ?T H E SLOPE OF THIS NEW CURVE WOULD EVIDENTLY BE THE ACCELERATION. I AM GLAD TO THINK THAT NOBODY HAS YET GIVEN A N A M ETO THE RATE OF CHANGE OF THE ACCELERATION.

T H E S Y M B O L S IN USE AREs AN D t FOR SPA CE AND TI ME ;VELOCITY v OR ~ , O R NEWTO N'S SY M B O L * ;civ

dv d' ^sACCELERATION -y OR , OR NEWTON'S I.dt dt-R AT E OF CHANG E OF ACCELERATION WOULD BE .

d-sNOTE THAT IS ONE SYMBOL, IT HAS NOTHING WHATEVER TO DO

WITH SUCH AN ALGEBRAIC EXPRESSION AS ~ . T H E SY M B O L IScl X t"SUPPOSED MERELY TO INDICATE THAT WE HAVE DIFFERENTIATED STWICE WITH REGARD TO THE TIME.

I HAVE STATED THAT THE SLOPE OF A CURVE MAY BE FOUND B YDRAWING A TANGENT TO THE CURVE, AND HENCE IT IS EASY TO FINDTHE ACCELERATION FROM THE VELOCITY CURVE.


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A C C E L E R A T I O N . 2519 Another way better than bydrawngtangents, isillustratedinthis Table :

Vfeet persecondor d s/St

accelerationtseconds sfeetVfeet persecondor d s/St

infeet persecondpersecond ord o/St

06 088014-74

07 2354 13-49 - 12508 370312-22

- 12709 4925

1095- 127

10 60209-66

- 12911 6986

8-35 - 13112 7821 7-04 - 13113 8525

In a newmechanism it was necessary for a certainpurpose to knowin everyposition of a point Awhat itsacceleration was, and to dothis I usuallyfindits velocityfirst. Askeleton drawng was made and the positions ofAmarked at the intervals of time t froma time takenas 0. In the table I give at each instant the distanceof A froma fixed point of measurement, and I call it s.If I gave the table for all the positions of A till it getsbackagainto its first position it wouldbe more instructive,but any student can make out such a table for himselffor some particular mechanism Thus for example, let sbe the distance of a piston fromthe end of its stroke.O course the all-accomplished mathematical engineer wllscorn to take the trouble. He knows a graphical rulefor doing this in the case of the piston of a steamengine. Yes, but does he know such a rule for every


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26 CALCULUS FOR ENGINEERS.possible mechanism ? Would it be worth while to seekfor such a graphical rule for every possible mechanism ?Here is the straightforward Engineers' common-sense wayof finding the acceleration at any point of any mechanism,and although it has not yet been tried except by myselfand my pupils, I venture to think that it will commenditself to practical men. For beginners it is invaluable.

Now the mass of the body whose centre moves like thepoint A, being m (the we ig h t of th e body in pounds atLondon, divided by 3 2 - 2 ) * , multiply the acceleration infeet per second per second which you find, by m, and youhave the force which is acting on the body increasing thevelocity. The force will be in pounds.

* I have given elsewhere my reasons for using in books intended forengineers, the units of force employed by all practical engineers. I haveused this system (which is, after all, a so-called absolute system, just asmuch as the c. o. s. system or the P o u n d a l system of many text books) fortwenty years, with students, and this is why their knowledge of mechanicsis not a mere book knowledge, something apart from their practical work,but fitting their practical work as a hand does a glove. One might as welltalk Choctaw in the shops as speak about what some people call theEnglish system, as if a system can be English which speaks of so manypoundals of force and so many foot-poundals of work. And yet these samephilosophers are astonished that practical engineers should have a contemptfor book theory. I venture to say that there is not one practical enginesin this country, who thinks in Poundals, although all the books have usedthese units for 30years.In Practical Dynamics one second is the unit of time, one foot is theunit ofspace, one pound (what is called the weight of 1 lb. in London) is theunit of force. To satisfy the College men who teach Eng ineers, I would saythat "The unit of Mass is that mass on which tbe force of 1 lb. produce*an acceleration of1 ft. per sec. per sec."We have no name for unit of mass, the Engineer never has to speakof the inertia of a body by itself. His instructions are " In all Dynamical

calculations, divide the weight of a body in lbs. by 32-2 and you have itsma ss in En gineer's un itsin those units which will give all your answers inthe units in which an Engineer talks." If you do notuse this system everyanswer you get out will need to be divided or multiplied by something beforeit is the language of the practical m an.Force in pounds is the space-ride at which work in foot-pounds is done,it is also the time-rate at which momentum is produced or destroyed.Example 1. A H a m m e r h e a d of 2 J lbs. moving with a velocity of 40 ft.per sec. is stopped in -001 sec. What is the average force of the blow?Here the mass being 2^^-32-2, or -0776, the momentum (momentum ismass x velocity) destroyed is 3-104 . Now force is momentum per sec. andhence the average force is 3-104-f--001 or 3104 lbs.Example2. Water in a jetflowswith the linear velocity of 20 ft. per sec,


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DIFFERENTIATION OF a x 2 . 272 0 . We considered the case of fall ing bodies in whichspace and t ime are connected by the law s = \gt 1, where gthe acceleration due to grav ity is 3 2 2 feet per second persecond at London. Bu t many other pairs of things are connected by similar laws and I will indicate them generally b y

y = ax 2 .L e t a particular value of a be taken, say a = J ^ . Now takex = Q, x=l, x = 2, x = ' 3, &c. and in every case calculate y.Plot the corresponding points on squared paper. Th eylie on a para bolic curve. A t any poin t on th e curv e, saywhere x = 3 , find th e slope of th e curv e ^1 call it , dothe same at = 4 , x = 2, &c. Draw a new curve, now, with thesame values of x bu t with ^ as the ordinate. Th is curveaxshows at a glance (by the height of its ordinate) what isth e slope of th e first curv e. I f you ink these curve s, letthe y curve be black and th e ~- curve be red. N otice tha tthe slope or at an y po int, is 2a m ultiplied by the of theaxpoint.

W e can inves tigate this algebraically. As before, foran y value of x calculate y. Now take a greater value of xwhich I shall call x + Bx and calculate the new y, calling i ty + Sy. We have theny+Sy = a (x + Sx)-

= a {x- + 2x. Sx + (Sx) 2} .S u b t r a c t i n g ; By = a [2x. 8x + (Bu)-} .Divide by Bx, ^| = 2ax + a . Bx.(relatively to the vessel from which it flows), the jet being 0-1 s


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28 C A L C U L U S FO R E N G I N E E R S .I m a g i n e 8* to get smaller and smaller without l imit anduse the symbol for the l imit ing value of |^, and we havedy~ = 2ax, a fact which is known to us already from oursquared paper* .

21. N ote t ha t when we repeat the process of differentiation3? xiwe state the result as -~ a u ( I the answer is 2a, You m u s tdx-become familiar with these symbols. I f y is a function ofx, ^ is the rate of change of y with regard to x; ( ~ is thera te of change of ~ with regard to x.CLOUOr, shor t ly ; is the d i f f e r e n t i a l c o e f f i c i e n t of -~with regard to x\ is the differential coefficient of y withregard to x. d? ii dyO r, a g ai n ; i n t e g r a t e and our answer is ; i n t e NT/ .grate and our answer is y.

You will , I hope, get quite familiar with these symbolsand ideas. I am only afraid that when we use other let tersthan x's and ys you may lose your familiarity.* Symbolically. L E T y= f (x)...(l), WHERE/(AS) STANDS FOR ANY EXPRESSIONCONTAINING x. TAKE ANY VAIUE OF x AND CALCULATE y. NOW TAKE A SLIGHTLYGREATER VALUE OF x SAY x + dx AND CALCULATE THE NEW y CALL IT y + dy

THEN y+ Sy=f{x + Sx) (2).SUBTRACT (1) FROM (2) AND DIVIDE BY dx.dy _f(x + dx)-f(x)dx dxWHAT WE MEAN BY ~ IS TNE LIMITING VALUE OF /J^_+Afi)/W a s S x JSdx dxMADE SMALLER AND SMALLER WITHOUT LIMIT. TH IS IS THE EXACT DEFINITION OF .dxIT IS QUITE EASY TO REMEMBER AND TO WRITE, AND THE MOST IGNORANT PERSON MAYGET FULL MARKS FOR AN ANSWER AT AN EXAMINATION. IT IS EASY TO SEE THAT THEDIFFERENTIAL COEFFICIENT OF af (.R) IS a TIMES THE DIFFERENTIAL COEFFICIENT OF / (x)AND ALSO THAT THE DIFFERENTIAL COEFFICIENT OF / (x) + F (x) IS THE SUM OF THE TWO

DIFFERENTIAL COEFFICIENTS,


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M O T I O N . 2 9

The differential coefficient ofy = a + bx + cxs,where a, b and c are constants, is^ = 0 + 6 + 2cr.ax

The integral of 0 + b + kx with regard to x is A + bx + -^fcf2,where A is any constant whatsoever.Similarly, the integral of b + kz with regard to z is

A + bz + \kz".The integral of b + kv with regard to v, is A + bv + ^kv".

IT IS QUITE EASY TO WORK OUT AS AN eseria.SE THAT IF y = ax3, THEN


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so C A L C U L U S F O R E N G I N E E R S .Againintegrate ands =c+vt + at'\ Againyouwll noticethat we addanunknownconstant, whenwe integrate. Someinformationmust be givenus tofindthe value of the constantc. Thus if s =swhen t 0, thiss is the value of c andsowe have the most complete statement of the motion

s=5 +vt +i|ffi (8).If (3) is differentiated we obtain (2) andif (2) is differentiatedwe obtain(1).

23 . We see here, then, that a s s o o n a s t h e s t u d e n t i sa b l e t o d i f f e r e n t i a t e a n d i n t e g r a t e he canwork the followng kinds of problem

I. If sis givenas anyfunctionof the time, differentiateandthe velocityat anyinstant is found differentiate againandthe accelerationis found

I I . If the accelerationis givenas some function of thetime, integrate and we findthe velocity; integrate againandwe findthe space passedthrough.

Observe that s instead of being mere distance may bethe angle described the motionbeingangular or rotational.Better thencall it 6 Then6or - , - is the angular velocityand6or is the angular acceleration

2 4 . E x e r c i s e s o n M o t i o n w i t h c o n s t a n t A c c e l e r a t i o n .1. The accelerationdue togravityis downwards andis

usuallycalledg g being32'2 feet per secondper secondatLondon. If a bodyat time 0 is thrown verticallyupwardswthavelocityof V feet per second where is it at the endof t seconds ? If s is measured up-wards, the accelerationis g and.5=Vt hgt~ (We assume that there is noresistance of the atmosphere andthat the true accelerationis gdownwards andconstant.)

Observe that v = V 0 -gt andthat v =0when V~gt =0or t= - . When this is the case find s. This gives thehighest point andthe time takentoreachit.

Wheniss=0again? What is the velocitythen?


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K I N E T I C E N E R G Y . 312. The bodyof Exercise 1has beengiven in addition

toits vertical velocity a horizontal velocityv which keepsconstant. If x is the horizontal distance of it away fromd?cc d$the originat the time t, -3-- 0 and-j- = ;


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Bv But our equations are only entirely true whenBs, Bt, &c ., are made smaller and sm aller without limit :H e nc e a s < ^ = my, or in words, "the differential coefficient ofdvE with regard to v is mv," if we integrate with regard to v,E = \mv'--\-c where c is some constant. L et E 0 when = 0 s o t h at c = 0 an d w e h av e E = Amu2.

Practise differentiation and integration using other lettertsthan x and y. In this case stands for our old I f weJ dv dxhad had = mx it might have been seen more easily thaty= hiu:2 + c, but you must escape from the swaddling bandsof x and y.

2 6 . Exercise . I f x is t h e e l o n g a t i o n o f a s p r i n gFwhen a force F is applied and if a; - , a representing thestiffness of the spring; F. Bx is the work done in elongatingthe spring through the small distance Bx. If i^is graduallyincreased from 0 to F and the elongation from 0 to x, whatstrain energy is stored in t he sp ring ?

The gain of energy from x to x + Bx is BE = F. Bx, orra ther - 7 = F= ax, hence E lax- + c. Now if E = 0 whenax zx 0, we see t ha t c = 0, so th at the energ y E stored is

It is worth noting that when a mass M is vibrating at theend of a spiral spring; when it is at the distance x from its

Otherw ise. Le t a small body of mass m and velocity vpass through the very small space Bs in the time Bt gainingvelocity Bv and let a force F be actin g upon it. Now&vF= m x acceleration or F = m and Bs = v . Bt so thatotF. Bs = mvBt ^ = m . vBv = BE,oti f BFJ stands for the increase in the kinetic energy of the bodyBE-r- = m . v.


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E L E C T R I C ! C I R C U I T . 3 3

position of equilibrium, the potential energy is \aa? and thekinet ic energy is \Mir o r t h e t o t a l e n e r g y is \M- + \ax>....Note that when a force F is required to produce anelongation or compression x in a rod, or a def lexion x in a beam,and i f F = ax where a is some constant, the energy stored upas strain energy or potential energy is \ax- or ^Fx.A l s o i f a T o r q u e T is required to produce a turnin gthrough the angle 6 in a shaft or spring or other structure,and if T = ad, the energy stored up as strain energy orpotential energy is had2 or }/T6. I f T is in pound-feet and 6is in radians, the answer is in foot-pounds.W o r k d o n e = F o r c e x d i s t a n c e , o r T o r q u e x a n g l e .27 . I f the student knows anything about e le ctr ic i ty lethim translate into ordinary language the improved Ohm's law

V = R C + L . dC /dt (1) .Observe that if R (Ohms) and L (Henries) remain con

stant, if C and are known to us, we know V, and if thelaw of V, a changin g voltage, is known you may see that theremust surely be some means of f inding C the changing current.Think of L as the back electromotive force in volts when thecurrent increases at the rate of 1 ampere per second.

I f the current in the primary o f a t r an s f o rm e r , and therefore the induction in the iron, did not alter, there would beno el ect rom oti ve force in th e secondary. In fact the E.M.F.in the secondary is, at any in stant, the n umb er of turns ofthe secondary multiplied by the rate at which the inductionchanges per second. R ate of increase of / per second is whatwe now call the differential coefficient of / with regard totime. Although L is constant only when there is no iron orelse because the induction is small , the correct formula beingdTV=liC+ iV --- (2) , i t is found tha t, practically, ( I ) withL constant is of nearly universal application. S ee Art. 183.

2 8 . If y = ax" and you wish to f ind ~- , I am afraid that


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I must assume that yo u know the Binomial Theoremwhich i s :

6It is easy to show b y multiplication that the Binomial

Theorem is true when n = 2 or 3 or 4 or 5, but when n = h or or any other fraction, and again, when n is negative, yo uhad better perhaps have faith in my assertion that th eBinomial Theorem can be proved.It is however well that yo u should see what it means byworking out a few examples. Illustrate i t with n = 2, thenii = 3, n = 4, &c, and verify b y multiplication. Again tr yn = 1, and if you want to see whether your series is correct,just recollect that (x + is - ^ an d divide 1 b y x + b inthe regular way by long division.

Le t us do with our new function of .r as we did with ax-.Here y = ax", y + By = a (x + Bx)' 1 a [x" + n . Bx . xn~x?z (n 1)+ ^ - (Bx)- xn~~- + terms involving higher powers of Bx).Now subtract and divide b y Bx and yo u will find

| = a | n . A - - + ^ ( ~ - , ) - ( & / ) ^ 4 - & c . JWe see now that as Bx is made smaller and smaller, in

the limit w e have only the first term left, all the othershaving in them Bx or (Bx)- or higher powers of Bx, and theymust all disappear in the limit, and hence,

= naxn ~ 1 (See Notes p. 150.")dxThus th e differential coefficient of x6 is Gx", of x-i it is

2Jkr1?, and of x~%- it is - gart.When we find the value of the differential coefficient of

any given function we are said to differentiate it. Whengiven ^ to find y we are said to integrate. T h e origin of


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}x l.dx. 35the words differential and integral neednot be consideredTheyare nowtechnical terms.

Dfferentiate ax 11 andwe find nax n~\Integrate nax n~^ andwe findax n + c. We always adda

constant whenwe integrate.Sometimes we write these, (ax 11) = m % " ' ~ 1 and

nax n~ l . dx = ax" .Observe that we write J before and dx after a function

whenwe wshtosaythat it is tobe integratedwth regardtox Boththe symbols are needed At present yououghtnot totrouble your headas towhythese particular sorts ofsymbol are used*.

You wll findpresentlythat it is not dfficult to learnhowtodifferentiate anyknownmathematical function Youwll learnthe process easily; but integration is aprocess ofguessing and however much practice we may have, experience onlyguides us ina process of guessing To someextent one may saythat differentiation is like multiplicationor raising anumber to the 5th power. Integration islike dvision or extractingthe 5th root. Happily for theengineer he onlyneeds a veryfewintegrals and these are

* Whenagreat number of thngs have tobe addedtogether inanengneers offices whenaclerkcalcuates the weight of eachlittle bt of acastingandadds themall up if the letterwindcates generallyanyof thelittle weights, w oftenuse the symbo Twtomeanthe sumof themall.Whenwe indcate the sumof aninfinte number of little quantities werepace the Greekletter s or 2bythe longEngishs orJ. It wll be seenpresentlythat Integrationmaybe regardedas findnga sumof ths kindThus ifyis the ordnate of acurve ; astripof areaisy.Sc andjy . dxmeans the sumof all suchstrips, or the whoe area Agan ifdmstands forasmall portionof the mass of abodyandris its dstance fromanaxis, thenr2. 8miscalledthe moment of inertiaofdmabou the axis, and2r2. 5morjr*. dmindcates the moment of inertiaof the whoe bodyabou the axis.O ifSVis asmall element of the voume of abodyandmis its mass perunt voume thenjr-m. dVis the bodys moment of inertia


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3 6 C A L C U L U S F O R E N G I N E E R S .well known. As for the res t, he can keep a good long list ofthem ready to refer to, hut he had bet ter practise w orkingthem out for himself.

Now one is not often asked to integrate nax" ~ l. It is toonicely arranged for one beforehand . One is usu ally asked tobxm+1integrate focm. ..(l). I know th at th e answer is - ' - _ . . . ( 2 ) .How do I prove this ? By differentiating (2) I obtain (1) , therefore I know th at (2 ) is the integ ral of (1 ). Only I ough t toadd a constant in (2) , any constant whatever, an arbitraryconstant as it is called, because the differential coefficient ofa con stant is 0. Stu de nt s ough t to work out severalexamples, integrating, say, a~, bit?, b.i:-, c u r 5 , cx-, aak W h e none has a list of differential coefficients it is not wise to usethem in the reversed way as if it were a list of integrals, forthings are seldom given so nicely arranged.For instance jia3 . d.v x* . But one seldom is asked to

integrate 4a;s, more likely it will be 3x s or o,c\ that is given.We now have a number of interesting results, but thislast one includes the others. T hu s ify = x" or y = x' i or y = x1 or y = x"*,we only have examples of y = and it is good for th estuden t to work them out as examples. Thu s

dydxI f ii = 1 this becomes lx" or 1. I f n = 0 it becomes Oar 1or 0. B u t we hardly need a new way of seein g th at if y is aconstant, its differential coefficient is 0. W e know that ify = a + bx 4- c./ 4 - ex3 + &c. + gx'\T hen ^ = 0 + 6 + 2c>; + Sex- + &c. 4 - wix n-\dx Jwith this knowledge we have the means of working quite

* I suppose a student to know tha t anything to the power 0 is un ity. Itis instruct ive to actual ly ca lculate V>y logari thms a high r oot of a ny num berto see how close to 1 the answe r com es. A high root mea ns a sm all power,the higher the root the more nearly does the power approach 0.


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I N T E G R A T I O N O F

half the problems supposed to be difficult, that come beforethe engineer.The two important things to remember now, are : I f

y = ax 11, then = naxn-' ; and i f = th endxy = ^ x^ + c,

where c is some constant, orf 1)bx" ' dx = - -;/'"+> + c.

J III + 1I must ask students to try to discover for themselves(/ui l lus t ra t ions o f the fac t tha t i f y = x' 1, th en /- = nx' 1" 1. I dodxnot give here such il lustrations as happened to suit myself ;they suited me because the y were my own discovery. Iwould sugge st this, however :T a k e y = x7' . L e t x 1 -02, calculate y by logarithms.Now let x = 1'03 and calculate y. Now divide the incrementof y by "01, which is the increment of x.Let the second x be 1 '021, and repeat the process.Let the second x b e 1 02 01 , and repe at the process.I t will be found th at -g'~ is approachin g th e tru e valu e of

^ which i s o (1 0 2 ) ' .dx Do this again when >/= x" ' 7 for example. A student neednot think that he is l ikely to waste time if he works forweeks in manufacturing numerical and graphical i l lustrationsfor himself . G et really familiar with the simp le idea thatdydxf y = xn then = nxtha t I ax*. dx = x* +1 + constant ;J s + 1that \av*. dv = ^ tf+' + constant.J -s + lPr act is e th is wi th ,s= '7 or \S or 1/1 or 5 or -8, and useOther letters than x or v.


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3 8 C A L C U L U S E N G I N E E R S .

vs . dv The answer isi s

j\/v-. dv or Jv%. dv. Answer, f t At ~* . d. Answer, 2tK dx or /x~. dx Here the rule fails to help us forx J

we get ~ whichis cc, andas we canalways subtract aninfinite constant our answer is reallyindetermnate. In ourwork for some time to come we needthis integral in onlyone case. Later, we shall prove that

dx = log x , and I 1 dx = log (x + a) ,X J X "4" 3*

andif y=logx =- and Idv=loirvJ dxx J r b

Ifp=ar, then fi =3a?:2.M dvl{v =m, then ^ " _dt30. If pv=Rt, where R is a constant. Work the

followngexercises. Find , if vis constant. Answer, ^.Find^ , if pis constant. Answer, .at pThe student knows alreadythat the three variables p V

andtare the pressure volume andabsolute temperature of a" dpg a s . It is toolonwtowrite ~ when vis constant." We

29 . E x e r c i s e s . Find the followng Integrals. Theconstants are not addeder , lx\ | V S . dv Answer, v.

1 . .


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P A R T I A L D I F F E R E N T I A T I O N . 3 9s h a l l use for t h i s the s y m b o l t h e b r a c k e t s in d i c a t i n g t h a tt h e v a r i a b l e not t h e r e m e n t i o n e d , is c o n s t a n t .F i n d A n s w e r , A s p = lit. v~l we h a v e = - Mv~\

F i n d the c o n t i n u e d p r o d u c t of the second, fifth, and t h i r d oft h e a b o v e a n s w e r s and m e d i t a t e u p o n th e f a c t t h a t

G e n e r a l l y we may say t h a t if u is a f u n c t i o n of two

t h e n we s h a l l use the s y m b o l (-7- ) to m e a n the d i f f e r e n t i a l coeff ic ient of u w i t h r e g a r d to x w h e n y is considered tob e c o n s t a n t .

T h e s e are s a i d to be partial differential coefficients.3 1 . H e r e is an e x c e l l e n t e x e r c i s e for s t u d e n t s : W r i t e out any f u n c t i o n of x and y; c a l l it 1 1 .F i n d . Now d i f f e r e n t i a t e t h i s w i t h r e g a r d to y,

a s s u m i n g t h a t x is c o n s t a n t . T h e s y m b o l for the r e s u l t isdhidy. dx'I t w i l l always be f o u n d t h a t one g e t s the s a m e a n s w e r i fo n e d i f f e r e n t i a t e s in the o t h e r o r d e r , t h a t is

'dt\ _ v_.dp) ~ R

v a r i a b l e s x and y, or as wo sayu = / 0 , y);


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= ^ + 0 + 2axy + hf,( ] ^ r 0 + 0 + 2ax + 2by.

Again, ( 1 = 0 + 3//- + oa? + 2 bxy,an d j~~ 7 = 0 + 0 + 2ax + 2by,ax. a ywhich is the same as before.A stude nt ought not to get t ired of doing this. U se othe rletters than x an d y, and work man y examples. T he factstated in (3) is of enormous importance in Thermodynamicsand other applicat ions of M athem atics to engineering. Aproof of it will be given later. T he student ought h ere toget familiar with th e imp ortance of what will then b eproved.

32. One other thing may be mentioned. S uppose wehave given us that u is a function of x an d y, and thatfdu , .

[dx ~ + - r + C1? ' J +Th en th e integral of this isu = lax 4 + by3x + \ox'y + \gx-y- +/( y),where f(y) is some arbitrary function of y. This is addedbecause we always add a constant in integration, and as y is

regarded as a constant in f inding (^yj ; v e a d d f(y), whichmay contain the constant y in all sorts of forms multipliedby constants.

33. To il lustrate the fact, sti l l improved, that if y = log x,then ~ ~ = ^ . A s tudent ought to tak e such va lues o f x as 3,3 '0 0 1 ,3 '0 0 2 , 8 '0 0 3 &c , f ind y in every case, divide incrementsO I\ ?/ by the corresponding incremen ts of x, and see if our ruleholds good.

Thus t ry u = a* + f + (,x-y + bxy-,fdu\
http://gx-y-/http://gx-y-/http://gx-y-/


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S T E A M A N D P O W E R . 41Note that when a mathematician writes l o g x he alwaysmeans the Napierian logarithm of x .C dt34. E x a m p l e of / y = log < 4-constant.It is proved in Thermodynamics that if in a heat engine the working stuff receives heat 1 1 at temperature t, and if t0 is the temperatureof the refrigerator, then the work done by a perfect heat engine woudbe tf.'-l-W^-*If one pound of water at t 0 is heated to t lt and wo assume thatthe heat received per degree is constant, being 1400 foot-lbs. ; what isthe work which a perfect heat engine woud give out in equivalencefor the total heat? Heat energy is to be expressed in foot-pounds.To raise the temperature from t to t + b t the heat is 14006Y in foot-lb.This stands for 1 1 in the above expression. Hence, for this heat wehave the equivalent work S i r = 14O0S? f/l - , or, rather,

d ] ] = 1 4 0 0 - 1 4 0 0 ^ .dt tHence 11= 140CM - 1400f log t + constant.Now H=0 when0 = 140f0 - 1400C0 log ?0 +constant,therefore the constant is known. Using this value wefindequivalentwork for the heat given from t a to ^=1400 (


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4 2 C A L C U L U S FOll ENGINEERS.3 5 . E xe rc i s e s . I t i s proved in Thermody namics whenice and water or water and steam are toge the r at the sametemperature, if ^ is the volume of unit mass of stuff in thehigher state and s0 is the volume of unit mass of stuff in thelower state . Th en

L = t (*, - * 0)dpdtwhere t is the absolute tem perature, b eing 27 4 -f 6 C , Lbeing the latent h eat in unit mass in foot-pounds. I f wetake L as the laten t heat of 1 lb. of stuff , and sx and ,s- arethe volumes in cub ic feet of 1 lb. of stuff , the formula isstill correct, p being in lb. per sip foot.

I . I n Ice-water , .s = -0174 7, s, = -016 02 at ==274 (corresponding to 0 C ) , p being 2116 lb. per sq. foot, andZ = 7 9 x 1 4 0 0 . H e nce $ = - 2 7 8 1 0 0 .dt

And hence the tem pe rature of me lting ice is less as thepressure increases; or pressure lowers the melting point ofice ; that is, induces towards me lting th e ice. O bserve thedp _^ , the melting point lowers at theuanti tat ive meaning of

rate of '001 of a degree for an increased pressure of 2 78 lb.per sq. foot or nearly 2 lb. per sq. inch.I I . W ate r S team. I t seems almost impossible tomeasure accurately by experiment, .s-j the volume in cubicfeet of one pound of steam at any temperature. s for wateris known. Cal culate s0 from the above formula, at a fewtemperatures having from Regnault 's experiments the fo l lowing table . 1 think that the f igures explain them selves.

tabsolutepressure inlb. per sq.inch

Vlb. persq. foot SpStassumeddpdt

Lin footpounds1 0 0 .174 14-70 2116-4 81-510 5 3 79 17-53 2 5 2 4 87-8 740 ,710 22-2694 7 4 0 , 7 1 0110 3 84 20-80 2 9 9 4


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L A T E N T H E A T . 4 3I t is h e r e a s s u m e d t h a t the v a l u e of dp/dt for 105" C. is

half the sum of 81'5 and 94. T h e m o r e c o r r e c t way ofp r o c e e d i n g w o u l d be to p l o t a g r e a t n u m b e r of v a l u e s ofSp/ht on s q u a r e d p a p e r and get dp/dt for 105 C. m o r ea c c u r a t e l y b y m e a n s of a c u r v e , fSj - s 0 for 105 c C. = 7 4 0 7 1 0 -r (379 x 8 7 ' 8 ) = 2 2 ' 2 6 . N o w$0 '016 for c o l d w a t e r and it is not w o r t h w h i l e m a k i n g anyc o r r e c t i o n for its w a r m t h . H e n c e we may t a k e s = 22'28w h i c h is s u f f i c i e n t l y n e a r l y the c o r r e c t a n s w e r for the p r e s e n tp u r p o s e .Example. F i n d Si for 275 V. from the f o l l o w i n g , L b e i n g

t "F . 248 257 266 275 284 293 302 jP 4152 4854 5652 ; 6551 7563 8698 9966

Example. If the formula for steam pressure, p a8h where a and bare known numbers, and 6 is the temperature measured from a certainzero which is known, is found to be a useful but incorrect formulafor representing Regnault's experimental results ; deduce a formula forthe volume s of ono poun d of steam. We. ha.ve. also the well knownformula for latent heat L = c-et, where t is the absolute temperatureand c and e are known numbers. H ence, as which is the same asa is bad'' ', ,h - = Ir - el) -r tbff-K

After subjecting an empirical formula to mathematical operationsit is wise to test the accuracy of the result on actual experimentalnumbers, as the formula represents facts only approximately, and thesmall and apparently insignificant terms in which it differs from fact,may become greatly magnified in the mathematical operations.3 6 . Study of Curves. W h e n the e q u a t i o n to a newcurve is g i v e n , the p r a c t i c a l m an o u g h t to r e l y f i r s t u p o n hispower of p l o t t i n g it u p o n s q u a r e d p a p e r .

d.yV e r y o f t e n , if we find or the s l o p e , e v e r y w h e r e , i tJ dx 1 Jgives us a g o o d d e a l of i n f o r m a t i o n .I f we are t o l d t h a t xL, yx is a p o i n t on a c u r v e , and we areasked to find the e q u a t i o n to the tangent t h e r e , we h a v esimply to find the s t r a i g h t l i n e w h i c h has the s a m e s l o p eas the c u r v e t h e r e and w h i c h p a s s e s t h r o u g h xt yx. T h enormal is the s t r a i g h t li n e w h i c h p a s s e s t h r o u g h x\, y andwhose s l o p e is m i n u s the r e c i p r o c a l of the s l o p e of the c u r v e

there. See A r t . 13.


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44 CALCULUS FOR ENGINEERS.P (fig. 8) is a point in a curve APB at which thetangent PS and the normal PQ are drawn. OX and OY are

the axes. OR = x, UP = y, tan P&K = -~ ; the distance SBctx jis called the s u b t a n g e n t; prove that it is equal to y -f- -^ .The distance HQ is called the subnormal ; it is evidentlyequal to y y . The length of the tangent PS will be foundto be yy

dy) the length of the normal PQ isdy

1 + ( d i e I ' T h e I r l t e r c e P t 0 5 is * - y ;Example 1. Find the length of the subtangent and subnormal of the Parabola y =

^ 1 = 2 m,Hence Subtangent = mx- - r - 2mx or \x.

Subnormal yx 2mx or Im-or.Example 2. Find the length of the subtangent of y = mx11,

v- = mnxn~i.axSubtangent = mxn - r - WTWT_1 xjn.


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C U R V E S . 45Example 3. F i n d of w h a t c u r v e th e s u b n o r m a l is c o n s t a n t

i n l e n g t h , dy dx 1y -j - = a or ~r- - y.J dx dy a JT h e i n t e g r a l of i y w i t h r e g a r d to y is x = -J - y 2 + a c o n

s t a n t &, and t h i s is the e q u a t i o n to the c u r v e , w h e r e b m ayh a v e any v a l u e . It is e v i d e n t l y one of a f a m i l y of p a r a b o l a s .( S e e Art . 9 w h e r e 's and y's are m e r e l y i n t e r c h a n g e d . )

Example 4. The p o i n t a; = 4, y = S is a p o i n t in the p a r a b o l a y = % xk F i n d the e q u a t i o n to the t a n g e n t t h e r e . T h es lop e is ^ = J, x | . f ~ i or, as * = 4 t h e r e , the s l o p e is J x or .T h e t a n g e n t is t h e n , y = m + To f ind m we h a v e y = 3w h e n x = 4 as t h i s p o i n t is in the t a n g e n t , or 3 = m -f-1 x 4,s o t h a t wi is an d th e t a n g e n t is y = 1^ + {;./.

Example 5. The p o i n t = 3 2 , y = 3 is e v i d e n t l y a p o i n t int h e c u r v e y = 2+ F i n d the e q u a t i o n to th e n o r m a l t h e r e .

T h e s l o p e of th e c u r v e t h e r e is " r = 7V1' ^ = t o o n l K ^ ^ n es lop e of the n o r m a l is m i n u s the r e c i p r o c a l of t h i s or 160.H e n c e the n o r m a l is y m 160-r. But i t p a s s e s t h r o u g ht h e p o i n t x = 32, y = 3 and h e n c e 3 m 100 x 32.

H e n c e m = 51 23 an d th e n o r m a l is y 5 1 2 3 16 0 .UExample 6. A t w h a t p o i n t in the c u r v e y = ax~n is t h e r et h e s l o p e 6? d,,-.- = ? ia* '~" _1 .T h e p o i n t is s u c h t h a t i ts x sa t i s f i es naar"" 1 b or,l

>4-l# = ^ - y j . K n o w i n g i ts x we k n o w i ts y f r om thee q u a t i o n to the c u r v e . It is e a s v to see and w e l l to r e m e m b e r t h a t if xt y is a p o i n t in a s t r a i g h t l i n e , and if the s l o p eo f the l i n e is b, t h e n the e q u a t i o n to the l i n e m o s t q u i c k l yw r i t t e n is


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4 6 CALCULUS FOR ENGINEERS.He nce the equation to the tangent to a curve at the

point x ly y1 on the curve isV ~ y = the ^ at the point.And the equation to the normal is

y~ -9- = the at the point.x x x ayExercise ^. Find the tangent to the curve x"Hj ' 1 = a, at

the point x\, yl on the curve. An swer, a? + - y= m + n.yiExercise 2. Fin d the normal to the same curve.

Answer, - (x - x\) - - (y - y,) = 0.yi * iExercise 3. Find the tangen t and normal to the parabolay" = 4


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M A X I M A A N D M I N I M A . 47

I t seems as i f x 6, giving the product 36, were the correctanswer. B u t if we want to be more exact, it is good to ge ta sheet of squared paper; call the product y and plot thecorresponding values of x an d y. The student ought to dothis himself .

Now it is readily seen that where y has a maximum or aminim um value, in all cases the slope of a curve is 0 . Fin dthen the point or points where dyjdx is 0.Thus i f a number a is divided into two parts, one of themx and the other a x, the product is y = x (a x) or ax x1,

and = a 2.c. Fin d where this is 0 . Evid entl y wheredx Jx2x = a or x = |a.T he practical man has no great diff iculty in any of hisproblems in f inding whether it is a maximum or a minimumwhich he has found. In thi s case, let a = 12. Then x = 6gives a product 36. Now if x = 5'999, the other part is 6 '001and the product i s 3 5 99 99 99 , so tha t x = 6 gives a greaterproduct than x = 5'999 or x= 6"001, and hence it is a maximum and not a minimum value which we have found. T hisis the only method that the student will be given of distinguishing a maximum from a minimum at so early a periodof his work.Example 2. D ivide a num ber a into two parts suchthat the sum of their squares is a min imum . I f x is onepart, a x is the other. T he question is then , if

y = x- + (a xf, when is y a minimum ?y = 2x- + a- 2ax,

d/u~-= 4x 2a, and this is 0 when x = ia.ax

Example 3 . Wh en is the sum of a num ber and i tsreciprocal a minim um ? Le t x be the number and y =x +W h e n i s y a minimum ?

The differential coefficient of - or x" 1 being x~" , we have= 1 , and this is 0 when x = 1 .


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F U E L U N AV O Y A G E . 49That is, whenis 83-c 32.c amaximum? Answer, When836'4=0, orxis about 13cubic feet.I amafraidtomake M Grover responsible for the above

result whichI have drawnfromhis experiments. Hs mostinterestingresult was, that of the above 13cubic feet it isverymuchbetter that only9or 10 should be air thanthatit shouldall be air.

Example b\ Prove that ax x? is amaximumwhenx = 1 a.

Example 7. Prove that x a? is amaximumwhenx = i V 3 .

Example


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5 0 C A L C U L U S I ' O T T E N G I N E E R S .

* ASSUMING THAT YOU KNOW THE RULE FOR THE DIFFERENTIATION OF A QUOTIENTUSUALLY LEARNT AT THE VERY BEGINNING OF ONE'S WORK IN THE CALCULUS, ANDWITHOUT ASSUMING a TO HE 0 AS ABOVE, WE HAVE(.r -v) ihx- a + bx3.

2bx"' 3bvx-=a (1).

velocity x so as to make the consumption of fuel a minimumfor a given distance vi. The velocity of the ship relativelyto the bank of the river is x - v, the time of the passage is- ' ~ t > a l J ^ therefore the fuel burnt during the passage isin (a + bx'")

x vObserve 1 hat a + bx* with proper values given to a an d bmay represent the total cost per hour of the steamer, in-eluding interest and depreciation on the cost of the vessel,besides wages and provisions.You cannot yet differentiate a quotient, so I will assumex"a 0, and the question reduces to th i s: when is ax, vminimum ? Now this is the same question a s: wh en is't .a maximum ? or when is ar s vx~ 3 a maximum ? T hex3differential coefficient is 2,r~ a-f 3iu'. Pu tti n g this equal

to 0 we find x = v, or th at th e speed of th e ship relatively tothe water is half as great again as that of th e current.Notice here as in all other cases of maximum and minimum tha t the engin eer ought not to be satisfied m erely withsuch an answer. ,r = | is undoubtedly the best velocity, itmakes x ::i(x v) a minim um. B ut suppose one runs at lessor more speed than this, docs it mak e much d

perry, john - the calculus for engineers

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