MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
539 
vii. General Circular Cubic .—§§ 111, 112. 
111. We have already seen that the equation 
ax 3 + by ' 3 + cz" + dw" = 0, 
represents a circular cubic when 
a b c , d 
3 + 
r l V V i 
(126) 
where r x , r 2 , r 3 , r 4 are the radii of the four principal circles. The curve also passes 
through the centre of each of these circles. 
By equation (104) we see that the equation to the asymptote (he., tangent at the 
1111 ' 
point —, —, —, —) is 
^'+-+-=0; 
(127) 
the equation to the tangent at the centre of the circle (x=0) being 
a — b a — c a—cl 
-y+ — z + — ir=o, 
r z r 3 r i 
which is clearly parallel to the asymptote. 
Hence the tangents to the curve at the centres of the principal circles are all 
parallel to the asymptote. 
112. As in the case of the general bicircular quartic, there will be four systems of 
bitangent oircles, and on each principal circle there will be four single foci. There are 
clearly no double tangents. And if one of the principal circles degenerates into a 
straight line, the asymptote is perpendicular to it. 
viii. Nodal Circular Cubic .—§ 113. 
113. The equation considered in § 95, 
ax" -f by 3 + cz 2 = 0, 
is a circular cubic, when 
a . b,c 
2 + ~ 2 +I — 0. 
N c" 
This curve is the inverse of a conic with respect to a point on the curve. The 
curve passes through the centres of the principal circles, the tangents being respec¬ 
tively, by equation (113), 
3 Z 2 
