MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
535 
100 . 
The foci corresponding to the principal circle (x—0) are given by 
and 
Hence we have 
if cz 2 4:ivz 
-A—-J- : 
7 0 I 
b—a os a 
0, 
y°= 4 ZW. 
o o 1 
y z z* 4&w 
c(b~ci ) cib c(b—a) 
( 117 ) 
Similarly the coordinates of the two foci on the circle y= 0, may be written down. 
101. From the form of equations (117), it appears that all curves given by the 
equation 
„ 2 „ o o 
a r .*■ 
a 2 + K 1 6 2 + /c 1 c 2 
are confocal with the curve 
x 2 if z 2 
a i+}.+-.= 0 ’ 
Subtracting, we have 
O 0 
ar r 
a 2 (ar + k) 5 2 (5 2 + k) 
Hence the circles whose coordinates are respectively proportional to 
— A o — • 
S b 2 ’ ’ 2 c 2 ’ 
-5- o ~~S . 
a 2 + /c’ 5 2 + /c’ ’ 2c 2 ’ 
must cut orthogonally, but these circles touch the curves at their common points; 
hence confocal curves cut orthogonally. 
Since we have a quadratic to determine k when (xyz) are given, through any point 
two curves can be drawn confocal with a given curve ; and two nodal circular cubics 
can also be drawn with the same node and confocal with a given nodal bicircular 
quartic. 
102. The equation 
ax~ + by~-\- (S— 0 
represents in general a nodal bicircular quartic, and by inverting with respect to the 
node, we see that it is the inverse curve of an ellipse or hyperbola, according as a and 
b have the same or opposite signs, with respect to some point in the same plane. 
Such a curve then has two principal circles, with two single foci on each : it has 
also four double tangents, two from the centre of each principal circle. 
If one of the principal circles becomes a straight line, it divides the curve 
