MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
623 
where ( xyzw ) are the squares of the tangents from any point P on the surface, to the 
four bitangent spheres. 
285. By § 260 we can draw a series of bitangent spheres belonging to each system 
orthogonally to any sphere, and the points of contact of each series lie on a sphere 
which cuts the principal sphere belonging to that series of bitangent spheres orthogo¬ 
nally, and may be called the polar sphere of the given sphere with respect to the 
cyelide. Now let (x, y, z ) be any three bitangent spheres cutting a fixed sphere S 
orthogonally, and let w be the corresponding polar sphere of S, the equation of the 
cyclide must take the form 
d 2 w 2 + cPx? + b 2 y 2 +c 3 z a — 2 beyz — 2cazx — 2abxy=0, 
and the corresponding focal quadric will be 
2 
d? be ca ah 
Thus the polar of the centre of w with respect to the quadric is the plane passing 
through the centres of x, y, z. 
As a particular case we may suppose S replaced by the centre of the principal 
sphere, so that (x, y, z) become double tangent planes ; the plane 8 = 0 is now at an 
infinite distance, and so we see that the centre of the polar of the centre of the prin¬ 
cipal sphere with respect to the cyclide is also the centre of the corresponding focal 
quadric. And also the asymptotic cone of the quadric cuts orthogonally the double 
tangent cone from the centre of the corresponding principal sphere. 
Normals to a Cyclide from any Point.— §§ 286, 287. 
286. The problem of drawing normals to a cyclide has been extensively discussed 
by Dakboux (‘ Sur une Classe remarquable de Courbes et de Surfaces Algebriques.’ 
Note XI.). To find the number of normals which can be drawn from any point he 
proceeds thus :— 
Let (grjt ,be any tangent sphere to the cy elide 
then since 
we have the equations 
and 
ax 2j rby 2 -\-cz 2 -\- dw 2 -\-ev 2 =0, 
r) f co 
W 
(a + k)x {b + Tc)y (c + k)z ( d+k)w (e + k)v 
^2 
-l -j_ J _1_ 
ci -\-Jc () ■ ]- /’ c*\~lc d + k 6 -f 7u 
; 0 5 
? 2 
<y 4 . sr * 
i-hZTT 7L=°- 
{a + kf ' (b + kf 1 (c + kp ' (d + kf ' (e + A :) 2 
