Prof. Cayley on a Quartic Surface. 21 



The planes in question will coincide with the plane of the 

 conic, if only 



(b 2 - a 2 ) (a 2 + 7 2 + h 2 ) + a 2 * 2 = 0, 



or, what is the same thing, 



b 2 u*--(a 2 -b 2 )y 2 =:b 2 {a 2 -b 2 )-, 



that is, if the point (a, 0, y) be situated on the hyperbola y = 0, 



x 2 z 2 

 -2 — ji — -jh = 1 • The hyperbola in question and the ellipse z — 0, 



x 2 y 2 



-2 + jo = 1, are, it is clear, conies in planes 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 

 conies of a system of confocal ellipsoids. 



The surface in the case in question, viz. when the parameters 

 a } b } a, /5 are connected by the equation 



"* 7 2 _t 



is in fact the " Cyclide " of Dupiu. It is to be noticed that 

 we have here 



(a-0COs(9) 2 + & 2 sin 2 (9 + 7 2 



= a 2 + y 2 + b 2 --2acicos0 + (a 2 -b 9 )cos ,2 0-, 



a 2 u 2 

 which, observing that a 2 + <y 2 -f b 2 is = -g — 75, gives 



(*-<zcos0) 2 + b 2 sin 2 +7 2 =A/« 2 ^P costf - y4^Tp) i 

 so that the radius of the variable sphere is 



Va 2 -b 2 cosd- 



act 



\/a 2 -b* 



If the variable sphere, instead of passing through the point 

 («, 0, y) on the hyperbola, be drawn so as to touch a sphere 

 of radius /, having its centre at the point in question, then the 

 radius of the variable sphere would be 



V^^cos0-^J^-/, 



which is in fact 



, aoJ 



= >/V-& 2 COS0- /-g— tt J 



V a 1 — b x 



