468 Mb MAXWELL ON THE 



t being the variable, by the change of which we pass from one position of the line to another. 

 This line always passes through the circle 



% = 0, a? 2 + y 2 = a 2 , 



and the straight lines % = c, a? = 0, 



and z = — c, y = 0, 

 which may therefore be taken as the directors of the surface. 



Taking two consecutive positions of this line, in which the values of t are t x , and t L + St, 

 we may find by the ordinary methods the equation to the shortest line between them, its length, 

 and the co-ordinates of the point in which it intersects the first line. 

 Calling the length $£ t 



i.,^ ac . « 



S? = , sin 2t^t, 



S VV + c 2 



and the co-ordinates of the point of intersection are 



oo = 2a cos 3 t, y =2a sin 3 t, z = — c cos 2t. 



The angle o9 between the consecutive lines is 



30 = . a St, 



v« ! + 1- 2 



The distance Sa between consecutive shortest lines is 



. 3a + 2c . . 



da = — j sin 2t dt, 



s/a* + e* 



and the angle $<p between these latter lines is 



30 = - r - c h. 



r x/aF+c* 



Hence if we suppose £, 6, a, (p, and t to vanish together, we shall have by integration 



r - 2 v5T? <1 - cos2 ' ) - 



V a + c" 



3a + 2c . 



(1 - cos 2t), 



^ 



2v/a 8 + 

 c 



y/a 2 + c' 



By bending the surface about its generating lines we alter the value of (p in any manner 

 without changing £, 9, or a. For instance, making <f> = 0, all the generating lines become 

 parallel to the same plane. Let this plane be that of my, then £ is the distance of a generating 

 line from that plane. The projections of the generating lines on the plane of xy will, by their 

 ultimate intersections, form a curve, the length of which is measured by <j, and the angle which 

 its tangent makes with the axis of x by 9, 9 and <y being connected by the equation 



3a + 2c 2 \/a' 



2 \Za" + c 2 



(l _ cos — 9), 



which shows the curve to be an epicycloid. 



