536 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
symmetrically; corresponding, in fact, to the case where a conic is inverted on a point 
in one of its axes. 
If the radii of both principal circles are infinite the centre of inversion is the centre 
of the conic : e.g., the lemniscate, whose equation would be of the form z 2 =a 2 (afi-|-y 2 ). 
iv. The Limagon .—§ 103. 
103. If one of the principal circles, in the last section, becomes a straight line, and 
one of the two foci coincide with the centre of the principal circle, the nodes at infinity 
become cusps. This case corresponds to inversion of a conic on a focus. 
Suppose that r 2 is infinite in § 95, then the condition that the curve should be a 
Limacon is, from equation (117), 
4r{ 2 c(a — l>)= 4e 2 ah — <Tc{a — h) ; 
which, since we must have 
1 4 
becomes 
4 ah=c(a — h) .(118) 
The double tangents perpendicular to the axis are given by 
A, 2 cv“ 4vp 
ch-b~ = ’ 
and 
— = 0 ; 
j’ x e e 
which equations are satisfied by \r 1 =ve=pe: so that there is only one double 
tangent. 
v. Bicircular Quartic having a Cusp. —§§ 104-109. 
104. The equation of the curve may be reduced (by § 83) to the form 
x 2 =2ayz. 
The system of reference being the principal circle (x=0) ; the circle {y= 0) passing 
through the cusp, and the other point in which the curve cuts the principal circle ; the 
cusp (-=0); and the other point [iv— 0) common to the curve and principal circle. 
Let i\, r 2 be the radii of the two circles, e the distance between their points of 
