112 TRANSFORMATION OF SURFACES BY BENDING. 



of the generating lines on the plane of xy will, by their ultimate intersections, 

 form a curve, the length of which is measured by <r, and the angle which its 

 tangent makes with the axis of x by 0, and or being connected by the equation 



3a' + 2c* /. 2ja* + c' \ 



=-=== 1 - cos - 0], 



2>/a' + c'\ / 



which shows the curve to be an epicycloid. 



The generating lines of the surface when bent into this form are therefore 

 tangents to a cylindrical surface on an epicycloidal base, touching that surface 

 along a curve which is always equally inclined to the plane of the base, the 

 tangents themselves being drawn parallel to the base. 



We may now consider the bending of the surface of revolution 



Putting r = /x' + y 1 , then the equation of the generating line is 



This is the well-known hypocycloid of four cusps. 



Let s be the length of the curve measured from the cusp in the axis of z, 

 then, 



wherefore, r = ()*c~*s*. 



Let 6 be the angle which the plane of any generating line makes with 

 that of xz, then s and determine the position of any point on the surface. 

 The length and breadth of an element of the surface will be 8s and r86. 



Now let the surface be bent in the manner formerly described, so that 

 becomes ff ', and r, r , when 



& = u.d and r f =-r, 

 /* 



then / = (t) l c- | /i- 1 * 1 



= (f)*c'-M, 

 provided c' = ^c. 



The equation between r' and s being of the same form as that between 

 r and a shows that the surface when bent is similar to the original surface, its 

 dimensions being multiplied by /*'. 



