74 
ON THE CYCLIDE. 
[569 
magnitude than — 7, it is a hyperbola having the axis of x for its transverse axis; 
and finally z — 7, it is the two-fold line x = 0. It is easy to see the forms of the 
cubic curves which are the sections by any planes x = const, or y — const. 
Fig. 3. 
The before-mentioned circles are curves of curvature of the surface; to verify this 
a posteriori, write 
U = z (z - /3) (z — 7) + (z — 7) y 2 + (z — /3) x 2 = 0 
for the equation of the surface; and put for shortness P = 3z 2 —2z (/3 + 7)+ /3y, P + a? + y 1 — L, 
so that d z U = P + a? + y 2 , =L. The differential equation for the curves of curvature is 
2x{z-$) , 2y (z — 7) , 
xdz + (z — /3) dx, ydz + (z — 7) dy, 
dx , dy , 
P + x 2 + y 2 
\P'dz + xdx + ydy 
dz 
or, say this is 
H = dx 2 .2xy (z — 7) — dy 2 .2xy (z — /3) + dz 2 .2xy (7 — /3) 
+ dzdy . x [— 2 (z — /3) (2z — /3) + L] 
+ dxdz.y[ 2 (z — 7) (2z — 7) — L] 
+ dxdy. [(7 — /3) P + (2 z — ¡3 — 7) (y 2 — a?)] = 0. 
But in virtue of the equation U = 0, we have identically 
{2 (z- /3) xdx + 2 (z -y)y dy + Ldz] x {~ JZTpydx + xdy+ ^ (z-y) 
+ il 
= (7-/3) js - x -y( z -7) dzdx -x(z-/3)dzdy+ (z-/3)(z-7) dxdy}.