624 
MR. R. LACHLAN - OH SYSTEMS OF CIRCLES AND SPHERES. 
Now suppose 
% V £ Ot) V I 
a+/u,a' /3 + /ji/3’ y + /xy' 8 + /x8' e + /u.e' 
then substituting and eliminating Jc, we have clearly an equation of the twelfth 
degree in /x ; hence we infer that twelve spheres can be drawn through the circle 
common to the spheres (a/3ySe) (a'fi'y'S'e) to touch the cyclide. 
We may deduce from this that through any straight line can be drawn twelve 
tangent planes to the cyclide, or a cyclide is in general of the twelfth class. 
And, again, taking 77 , 4 , w, proportional to £C+-, y-f 7 , wJ r 7 , v-\-~ 
T l r 2 r 3 T i T o 
respectively, we see that twelve spheres can be drawn to touch the cyclide, having 
their centres coincident with the point (xyzwv). Hence twelve normals can be drawn 
to the cyclide from any point. 
287. In § 257, equations (279) give the normal at the point {x y'z'iv'v) of the 
cyclide 
aar+ by z + cz 2, + dw 2 +e v z = 0, 
and we infer that the feet of the normals from the point ( xy'zw'v') lie on the cubic- 
surfaces : 
ax, 
CZ, 
dw, 
ev 
X, 
y> 
Z, 
w, 
V 
/ 
y\ 
/ 
IV, 
f 
V 
1 
1 
1 
1 
1 
V 
r 2 
V 
5 
’4 
r o 
(309) 
Now, xp= 0 being the equation of the absolute referred to any system of spheres, 
<j>i, < 1 * 2 , 4 > 2 , any three cyclides ; let us form the discriminating quintic of 
\<py + + p\jj=0 ; 
then we can so choose the ratios X : /x : v : p that three of the roots of this quintic 
shall be zero ; and thus the equation will represent two spheres. To determine the 
ratios -, - we obtain three equations of the fifth, fourth, and third degrees respec- 
P P P 
tively. Hence through the common points of three cyclides a pair of spheres can be 
drawn in sixty ways. 
Let us suppose now that certain of the feet of the twelve normals from the point 
(xy'z'iv'v) lie on the sphere 
gx-\-r)y-\-& + (oiv-\-z3V=0, 
then the rest must lie on another sphere which we may denote by 
ctx-\- fiy -\-yz-\- S?r-|-er= 0, 
