MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
621 
280. If 
a 2/j t d 
Y + ITT +72 —^ 
r i r 2 r s e" 
the surface represented by the equation 
ax 2 + 2 hyz -f die 2 = 0, 
will be a cubic surface, which passes through the centre of the principal sphere, the 
tangent at which point is parallel to the plane 
ax , hz liy , dw 
+ -+-+—=o, 
r \ r 2 r 3 e 
and therefore passes through the line on the surface at infinity. 
Chapter YII. —Miscellaneous Theorems. 
Equation of a Cyclide referred to Four Spheres Orthogonal to a Principal Sphere .— 
§§ 281-285. 
281. Let the system of spheres [xyzwv) be such that v—0 is a principal sphere of a 
cyclide ; then its equation must take the form 
v 2 -\- <j>[xyzw) = 0, 
and if v be orthogonal to the system ( xyzw ), then the equation of the absolute must 
also be of the form 
v 2 fixyzw)— 0. 
Hence, by subtraction, we see that the equation of a cyclide can always be written 
in the form 
ax 2 -f- by 1 + cz 2 -f- dw 2 -\- 2 fyz + 2gzx + 2hxy +2 lxw-\- 2myiv + 2nziv = 0 ; . (305) 
the system of reference being four spheres, points or planes orthogonal to a 
principal sphere. 
Now, the equation of any sphere orthogonal to this same principal sphere is of the 
form 
uxffiy+yzf- hiv— 0,.(306) 
where, by § 211, equation (222), a., y, S must be proportional to the tetrahedral 
coordinates of the centre of the sphere, referred to the tetrahedron formed by the 
centres of [xyzw), provided that [xyzw) the coordinates of any point are the powers of 
the point with respect to the system. 
