[569 
569] 
ON THE CYCLIDE. 
75 
for its transverse axis; 
-o see the forms of the 
y = const. 
he surface; to verify this 
= 0 
8 +7)+ /3y, P + a? + y 2 = L, 
curves of curvature is 
y 2 
= 0, 
■hydy 
(7-/3) 
). 
+ 
xy (7 ~ ¡3) 
(z-$)(e-7) 
Hence in virtue of the equations U = 0, dU = 0 the equation 0 = 0 becomes 
xydz 2 — y(z — 7) dzdx — x (z — ¡3) dzdy + (z — ¡3) (z — 7) dx dy = 0, 
that is, 
[xdz — (z — 7) dx) [ydz — (z — /3) dy) = 0, 
whence either x — G (z — 7) = 0 or y — O' (z — /3) — 0; viz. the section of the surface by 
a plane of either series (which section is a circle) is a curve of curvature of the surface. 
The equation of the cyclide can be elegantly expressed in terms of the ellipsoidal 
coordinates (X, ¡x, v) of a point (x, y, z); viz. writing for shortness a. = b 2 — c 2 , /3 = c 2 — a 3 , 
7 = a 2 — b 2 , the coordinates (X, /x, v) are such that 
— ¡3yx 2 = (a 2 + X) (a 2 + fx) (a 2 + v), 
— rytxy 2 = (b- + X) (b 2 + y) (b 2 + V), 
— afiz 2 = (c 2 + X) (c 2 + /x) (c 2 + y), 
(see Roberts, Comptes Rendus, t. liii. (Dec., 1861), p. 1119), whence 
x 2 + y 2 + z 2 = a 2 + b 2 + c 2 + \ + fx + v, 
(b 2 + c 2 ) x 2 + (c 2 + a 2 ) y 2 + (a 2 + b 2 ) z 2 = b 2 c 2 + c 2 a 2 + a 2 6 2 - fxv — v\ — \/x. 
The equation of the cyclide then is 
\/ (as + X) + \Z(a~ + /x) + V (a 2 + v) = V (S). 
In fact, starting from this equation and rationalising, we have 
(3tt 2 + X + fx + v — 8) 2 = 4 [\/[(a 2 + fx) (a 2 + v)) + \/((ft 2 + v) (ft 2 + X)} + V((ft 2 + X) (ft 2 + /ft)}] 2 
= 4 [3ft 4 + 2ft 2 (X + fx + v) + fxv + v\ + X/a + 2 VK« 2 + V) (& 2 + /ft) (ft 2 + y)} */(&)], 
which, substituting for 
X •+■ /ft -f- v, /xv -f- v\ -p X/ft and \/{(ft" ~P X) (ft 2 -p /a) (ft" -p z/)| 
their values, is 
(x 2 + y 2 + z 2 + 7 — /3 — S) 2 = 4 {(7 — /3) # 2 — /Qy 2 + 72 2 — /87 — 2x\/(— /87S)}, 
or, writing —17 2 , |y8 2 , ¿S 2 in place of /3, 7, 8 respectively, this is 
(x 2 + y 2 + z 2 + \y 2 + 1/3 2 — |S 2 ) 2 = (7 2 + /3 2 ) x 2 + (3 2 z 2 + y-y 2 + j/3 2 7 2 + /3<y8x, 
which agrees with a foregoing form of the equation. 
The generating spheres of the cyclide cut at right angles each of a series of 
spheres; viz. each of these spheres passes through one and the same circle in the 
plane of, and having double contact with, the conic which contains the centres of the 
generating spheres; the centres of the orthotomic spheres being consequently in a line 
meeting an axis, and at right angles to the plane of the conic in question. Or, what 
is the same thing, starting with a conic, and a sphere having double contact therewith, 
the cyclide is the envelope of a variable sphere having its centre on the conic and 
cutting at right angles the fixed sphere.* 
* I am indebted for this mode of generation of a Cyclide to the researches of Mr Casey. 
10—2 
izdy+ (z — /3) (z — 7) dxdy).