MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
613 
has the same principal spheres and the same focal carves as the surface 
+-+*=o • 
2 ' /32 I Q i v.2 I o w 5 
a- 7 O 6" 
and subtracting, we have 
™2 
a 3 (a, 3 + /c) / 3 ' 2 (/ 3 2 + k) 7 2 (7 2 + /c) t 2 (t 2 + /c) e 2 (e 2 + /c) 
= 0 , 
and therefore the spheres whose coordinates are respectively proportional to 
and 
x y z w v 
OS 02’ 2’ J.2S O 5 
or p“ 7 o e~ 
x y z w v 
0 i ? 02 E 5 0, , 5 V 0 . 5 2 i 5 
a" + /c p +k 7" + a: cr + /c e + /c 
cut orthogonally ; but these spheres evidently touch the surfaces at a common point; 
hence, confocal cyelides cut orthogonally. And since the above equation (282) is a 
cubic in k, it follows that through any point can be drawn three cyclicles confocal with 
a given cyclide, and these surfaces cut orthogonally. 
Or, again, let us determine k so that (282) represents a cubic cyclide ; then we see 
that three cubic cyclides can, in general, be drawn having the same focal curves as a 
given quartic cyclide, and these cubic cyclides cut orthogonally. Or, if the given 
cyclide is a cubic cyclide, three quartic cyclides can be drawn with the same focal 
curves through any point; and two other cubic cyclides can also be drawn with the 
same focal curves. 
260. Let (^y^wnr) be any sphere S ; this will cut orthogonally one of the bitangent 
spheres at the point (xy'zw'v') of the surface 
ax 2 + 6y 3 +C2 3 +dvr + ev 2 — 0, 
if 
(a—e)x'£-\-(b—e)y f y J r (c—e)z't,-\-{d—e)w'co=0. 
Hence, given any sphere S, a series of bitangent spheres belonging to any system can 
be drawn, cutting S orthogonally, their points of contact lying on the curve in which 
the cyclide is cut by the sphere 
(a—e)£x-{-(b — e)yy-\-(c — e)l,z-\-{d— e)uw=Q, 
which may be called the polar sphere of S with respect to the cyclide. 
There are five such polar spheres for any sphere S, each cutting one of the principal 
spheres orthogonally, and each one clearly intersects S in points lying on the sphere 
a^x-\-byy-\-c'Qz-\-d(uw-\-ezsv ~0 ; 
i.e. s the five polar spheres of any sphere have with S a common radical plane. 
