A SURFACE HAVING ONLY A SINGLE SIDE.* [ xl)bt :xl)u~I ,]

BY

CARL HERING, D.$~I~, Member of the Institute.

IN the August, I918, issue of this JOURNAL, page 233, the writer describ.ed an odd form of surface which differs from the

~i A 3'5 ,:A '~ .$ i

FI[3 14

c I FIG 14 a FIG 15

S , 'I ::S ~4 4- " , 2 ~,

3:A

usual surfaces in having ontv one side, and in a mathematical sense must therefore be considered non-polar o,r unipola,r.

Since then the writer has made a further investigation of t'he

----*Con]municated b y t h e A u t h o r _ . . . . . . . . . .

6.,.

628 CARL ]~{EI¢ING. [J. F. I.

algebraic equation of this surface which was added to the former article just before going to press, and has endeavored to give a bet- te.r concel~ti:on of this surface by mea'ns of someadditional drawillgs.

It is difficult to show this peculiarly curved surface by means of flat drawings other than stereoscopic views. Nor can it" be shown, by a shaded drawing as though made of cardboard, as it is in a sense hollow, or more correctly, doubly ,hollow; it should b.e shown as though 'transparent. The accompanying diagrams, Figs. I4, 15, I6 and I7, were therefore drawn accurately to scale sho,wing it in the form o0~ a succession of lines or rods representing its rectilinear elements in their successive positions in developing this surface. This gives the desired effect of transparency, and by making use of the conventional s,hading and the knobs at the ends, and by showing the self-intersection, the desired stereo- scopic effect is at least approached.

The two slanting, neighbo.ring, trum.pet-shaped holes or pas- sages through it, which in a sense make it doubly hollow, unfor- tunately cannot be sho.~'n as openings in any of these drawings, each of which is taken perpendicularly to some one of the four cardinal positions of the elements. These passages can be seen best in Fig. z6 when one conceives that the row of knobs through the middle represents a series of elements f.orming a partition which divides the tubular interior into. two ob,lique passages.

In all of these figures the four cardinal positions, I, -% 3 and 4 of Fig. I, are marked with their respective lmmbers. AA represents the central axis and S~S" the line along which this sur- face interests itself. The lengths of the elements were made equal to four times the diameter of .the circular directrix, which is shown in Fig. 17; beyond that distance there is notifing more of interest.

Fig. 14, corresponding to. the former Fig. 4, is a view in line with position I of Fig. I. taken from the right, while Fig. 16, corresponding to the former Fig. 6, is the same taken from the left. Fig. 15, corresponding to the former Fig. 5, is taken in line wit,h the plane of 2 and 4 (which two positions coin- cide in this view), hence is a view o,f Fig. 14 from the left, or of Fig. 56 from the right; it shows the two trumpet-shaped and the two fan-shaped parts. Fig. 17 is the top view of Fig. 15 in line with the axis ; eac,h of the positions ha this view necessarily represents two diametrically opp.osite elements which therefore

Nov.,1918.l ~\ SURFACF. t-lAVING .\ SIN(iLE SIDE. 629

partially overlap.; the knobs indicate the ends of these super- imposed elemen¢s.

F r o m the: group of rectilinear cross sections, Figs. 8 to ] 3 in the fo rmer arti.cle, one was over looked: it is the ver,tical sec- tion through the position .3 and the line of intersectio.rl~5", hence perpendicular to position ] : it is a cross section of Fig. ]4 parallel t~ the plane of the paper and slightly to the. rear of it ; it is shown on a reduced scale in t:he supplementary Fig. ]4a. It is one of the simplest of these cross sections.

In. the algebraic equaiion given in the fo rmer article there is an e r ro r in two of .the signs: t'he. sign before r3,~ should be - and not +. while the one a f te r 3,~ should he + ~nd not - . Dr. C. P. Steinmetz subsequemly reduced it to the fol lowing form :

~' (y+~) ~--5' ( ,+a~ %2.r ),.q-~) ( r+x) =o

The wri ter has reduced it to the following, which he believes to 1)e its simplest f o rm:

3' (.r"--Y"@ c~-r'~) +2~: (.v'a+y2+r.r) 0

In these the geometr ic centre of the surface is a.t the origin of the system of co(Srdinates, and the circle of radius r is in the plane of X and Y; hence the axis A in the figures is the axis of co- /;rdinates Z ; position i lies in the positive, direction of X. Both o.f the angular mo.tions are assumed to be clockwise, as shown in Fig. I ; w h e n one is clockwise and the other anti-clock- wise Ne only change in the latter equation is that the sign before the 2c is negative.

InserLting in this equation .r = - r gives y - - ~ , which proves mathematical ly that the self-intersection of this curved surface is a r ight l ine lying in a plane perpendicular to position ~ and through po.sition 3: and that it makes an angle Of 45 ~ with the horizontal or vertical, as shown in Fig. I4a. The result y - - ~ ' a]so shows that it lies in t'~he qugdrants b c, -3 ' , and + y, - ~ , all of which will be seen to be confirmed in the drawings by the line SS, and was t]rs,t noticed bv the wri ter in the models.

In all the above the clockwise direction was assumed for both of the angtllar mo,tions, as described in connection with Fig. ] : if both directions are anti-clockwise the results will, of course, be the same as befo.re: but if one is clockwise and the ,~.ther anti-clockwise this line of self-intersection S S will lie in the other pair of quadrants, that is, will b~ at r ight angles to its position in Fig. I4a, and its equation will then be y = ~, as the

630 CARL HERING. [J. F. I.

sign before 2~ in the above equation is then - ins~tead of +. These two. forms of this surface bear the same relations to each other as a right- and a left-handed screw thread.

As an additional proof of this curious rectiliear self-inter- section, the writer started with the assumption that it was this 45 " right line and developed the equation of a surface generated .by the movement of a right line so that it always touches: (a ) a circle, (b ) the axis of tile circle, and (c ) a right line making an angle of 45 ° wifll the plane of the. circle and lying in a plane tangential to the circle. The equation thus obtained was identical with the second one gi~,en above. The same surface may there- fore Ee correctly defined in either w a y .

TILe above equatio,ns were furthermore tested by deducing from them l:he cross sections, Figs. 8, 9 and ~o. Making c = o gives the equation of the section Fig. 8. Making 3,= o gives the equation of the section Fig. 9. Making :r = o gives the equation o.f the section Fig. IO.

Using polar co/Srdinates in tile plane of the circular direc- trix (the l)lane of X and l ' , see Fig. I7), and letting a be the angle of the I)rojection of an element on that plane measured clockwise, R the radius vector of the projection of any point of that element, and ~ as before the vertical distance o[ ~ that point a,bove or below this plane, then the equation at the end of the fo,rmer article (after correcting the signs) reduces to the much simpler forms :

z=(r--R) tan ~a

z=(R--r) co~ a--(R--r) cosecant a

from which the value of s o.f that point may be easily- determined. The first expression is the more convenient for arithmetical cal- culations, care being given to. the signs and to the directions of ro.tation. The second expression gives a completely graphical and very simple solution, for bv striking an arc through the point in question with a radius of R - r, using as a centre t h e point of intersection o:f the radius vector R with the circular directrix, and drawing the necessary co.tangen.t, the quantities ( R - r) cot a and ( R - r ) coscca,~t a., are given graphically, and as the lines representing them intersect, their difference, namely the desired value of ~0, is readily measured off. "Descriptive geom- etry," so-called, provides another completely graphical solution.