538 MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
These can only coincide if 
2 a_ 1 
_ 2 5 
er 2 r i 
in which case the curve is a circular cubic. 
108. Taking k = 0 in (119), we have a series of tangent circles passing through the 
cusp ; to find the tangents from the cusp to the curve we shall have 
X_ P- — 2 p 
whence 
where 
x —az' —ay'* 
Cl\~ ifJLp -0 \ 
X 
/A Zp 
- + -L- =0. 
r, r : 
The angle between these is given by 
tan </>: 
,1 2a1 * 
‘ i er i 
a 2 
e r n 
109. The focus of the curve is given by 
( y—azf— &wz=y 2 , 
whence 
8w 
(124) 
vi. The Cavdioid. —§ 110. 
110. If the radius r 1 of the principal circle of the curve 
cc 2 =2 ayz 
become infinite, the curve is symmetrical with respect to the axis; and if the single 
focus on this axis is at an infinite distance, the curve has cusps at infinity, and is 
called the cardioid; being the inverse of a parabola with respect to its focus. 
Referring to equation (124), we see that the condition is 
«= 4 ; 
so that the equation to a cardioid is 
x~— 8yz. 
(125) 
