542 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
and the corresponding focal conic is 
0 /OO o 
“~+§+r=o. 
a b c 
And we see that the focal conic corresponding to one principal circle is self-conjugate 
with respect to the triangle formed bj the centres of the other three. 
118. Or, again, in the case of a nodal curve; let (1, 2, 3) denote the other principal 
circle, and its two points of intersection with (4); the equation of the curve is of the 
form 
ax ' 2 d- % 2 + 2 fyz = 0; 
and the corresponding focal conic 
/ 3 « 2 — dby 2 + 2afl3y = 0, 
which also passes through the node. 
119. If the system of reference be three bitangent circles, the equation to the curve 
must be of the form 
a\Zx-\-b\/y-\-c\/z — Q ; .(131) 
and in that case the focal conic is 
O 7 0 0 
or , (r . c~ 
+;, + ~ — b. 
a h 7 
In particular we see that if A, B, C be three foci on the same principal circle of the 
quartic, or cubic, and P any point on the curve, we must have 
a.AP+ABP+e.CP=0.(132) 
120. Suppose that the curve is a Cartesian, having cusps at infinity, then the focal 
conics become circles. 
It follows that equation (131) will represent a Cartesian if a, b, c are proportional 
to the sides of the triangle formed by the centres of the three bitangent circles (xyz). 
Thus we have the theorem that, the sum of the products of the tangents, from any 
point on a Cartesian to any three bitangent circles of the same system, into the 
corresponding sides of the triangle, formed by the centres of the circles, is zero. 
121. Let the circles (2, 3) be any two bitangent circles, and let (1) be the circle 
passing through their four points of contact with the curve ; then the equation of the 
quartic must take the form 
x z =2fyz, 
and then the focal conic is 
fa*=2/3y. 
If (y, z) be foci, ( x ) might be called their directrix; and we see that the product of 
the distances of any point on a bicircular quartic from two foci on the same principal 
circle, is proportional to the square of the tangent from the point to their directrix. 
