360] 
NOTE ON A QUARTIC SURFACE. 
467 
at right angles to each other, having the transverse axes coincident in direction, and 
being such that each curve passes through the foci of the other curve; or, what is 
the same thing, they are a pair of focal conics of a system of confocal ellipsoids. 
The surface in the case in question, viz. when the parameters a, b, a, /3 are con 
nected by the equation 
is in fact the “ Cyclide ” of Dupin. It is to be noticed that we have here 
(a —a cos 6) 2 + b 2 sin 2 6 + y 2 = a 2 + <y 2 + b 2 — 2aa cos 6 + (a 2 — b 2 ) cos 2 6; 
= Va 2 — b 2 cos 0 7 
V a 2 — b 2 
If the variable sphere, instead of passing through the point (a, 0, 7) on the hyperbola, 
be drawn so as to touch a sphere of radius l, having its centre at the point in 
question, then the radius of the variable sphere would be 
= Va 2 — b 2 cos 6 — l, 
Va 2 — b 2 
which is in fact 
= Va 2 — b 2 cos 6 —, aa - , 
va 2 —b 2 
if only a' = a H— ; hence if 7' be the corresponding value of 7, the variable 
sphere passes through the point (a0, y') on the hyperbola, and the envelope is still 
a cyclide. The cyclide as derived from the foregoing investigation is thus the envelope 
of a sphere having its centre on the ellipse, and touching a fixed sphere having its 
centre on the hyperbola. It also appears that there are, having their centres on the 
hyperbola, an infinite series of spheres each touched by the spheres which have their 
centre on the ellipse; if, instead of one of these spheres we take any four of them, 
this will imply that the centre of the variable sphere is on the ellipse, and it is thus 
seen that the cyclide as obtained above is identical with the cyclide according to the 
original definition, viz. as the envelope of a sphere touching four given spheres. 
Cambridge, December 5, 1864. 
59—2