540 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
(b-a) 
- - -x — 
A 
2 a + c a 
—■—z — 2-w=0 
e e 
2h + c o h n 
—z — 2-U'—0 
e e 
which are parallel to the straight line 
ax 
r i 
+ ^+f=0, 
r„ e 
which is the line joining the node to the point in which the line at infinity cuts the 
curve, so that these three lines are parallel to the asymptote. 
As in the case of the nodal bicircular quartic there will be two single foci on each 
principal circle, and two corresponding systems of bitangent circles. 
114. The equation 
ix. Cuspidal Circular Cubic .—§ 114. 
x 2 = 2 ayz 
represents a circular cubic, having the point z —0 for a cusp, when 
i\ being the radius of its principal circle, r 3 that of a circle cutting this orthogonally, 
and passing through the cusp, and the other point common to the curve and its 
principal circle. 
The curve clearly passes through the centre of its principal circle, the tangent at 
the point being 
which is parallel to 
Toy 
ez 
l-w 
y L7+7 — 57i“ — 0 
x ay az 
? 'l e r 2 
the line joining the cusp to the third point in which the curve cuts the line at 
infinity; hence these are parallel to the asymptote. 
The curve has one syttem of bitangent circles, and one focus which lies on the 
principal circle. 
