TRANSFORMATION OF SURFACES BY BENDING. 469 



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 



+ y 2 + %% = ci 



Putting r = y/ai" + y'\ then the equation of the generating line is 



ri + *3 = ci 

 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 #, then, 



s = - c4 f» 



©-' 



wherefore, r= I-) c~$sl. 



Let 6 be the angle which the plane of any generating line makes with that of xz, then a 

 and determine the position of any point on the surface. The length and breadth of an 

 element of the surface will be & and r$9. 



Now let the surface be bent in the manner formerly described, so that 9 becomes 0, and 

 r, r ', when 



ff = fi.6 and r'= - r, 



2\i 



then r = i-J c~^/m~ 1 s^ 



provided c = (i*c. 



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

 the surface when bent is similar to the original surface, its dimensions being multiplied by /x*. 



This, however, is true only for one half of the surface when bent. The other half is pre- 

 cisely symmetrical, but belongs to a surface which is not continuous with the first. 



The surface in its original form is divided by the plane of xy into two parts which meet 

 in that plane, forming a kind of cuspidal edge of a circular form which Hmits the possible 

 value of 8 and r. 



After being bent, the surface still consists of the same two parts, but the edge in which 



they meet is no longer of the cuspidal form, but has a finite angle =2 cos -1 — , and the two 

 sheets of the surface become parts of two different surfaces which meet but are not continuous. 



J. C. MAXWELL. 



