532 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
This circle intersects S in points lying on the circle 
a£x + brjy -|- c (z+daw = 0. 
Hence, given any circle S, four pairs of bitangent circles can be drawn to any 
bicircular quartic, cutting S orthogonally; and their points of contact lie on four 
circles, which have with S a common radical axis. 
ii. Cartesian Oval .—§§ 93, 94. 
93. If one of the principal circles has its radius infinite, the curve will be symme¬ 
trical with respect to the axis, which will pass through the centres of the other three 
principal circles. If the foci which lie on this axis coincide with these centres, the 
curve must have cusps at the circular points at infinity. Let us suppose the circle, 
whose radius was r 4 , in § 86, to become the axis ; then by § 90, the coordinates of the 
foci on the axis will be 
( b—c){ct — d) (c—a)(b—d) ( ct—b)(c—cl )" 
If these points are the centres of the principal circles we must have 
r^(b—c)(a—d)=r£{c—a)(b—d)=r^(a—b)(c—d ); 
which is equivalent to only one relation between the coefficients, viz.:— 
1 11 111 
a — d b — d r 3 2 c — d oy ’ 
since 
? A+Vf- ? A=°- 
7 1 r 2 7 3 
Again the double tangents at right angles to the axis are given by 
±- + i*: + -A =0 
a—d'b—d'c—d ’ 
-+~+- = 0 ; 
n r :s 
which are clearly satisfied by taking 
. (Ill) 
Xr 1 =/tr s =v?’ s ; 
thus one of them coincides with the line at infinity, and so there is but one proper 
double tangent. 
