12 
[488 
488. 
NOTE ON A RELATION BETWEEN TWO CIRCLES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 82, 83.] 
Consider any two circles 0, Q; and let AG, BD, A'D', B'G' be the common 
tangents touching the circles in the points A, A', B, B', C, C', D, D': the locus of a 
point P such that the pairs of tangents from it to the two circles respectively form 
a harmonic pencil, is a conic through the 8 points A, A', B, B', G, G', D, D'; but 
this conic may break up into two lines, viz. if (as in the figure) the points A, B', D', D 
are in a line, then the points G, G', A', B will be in a symmetrically situated line, 
and the conic breaks up into this pair of lines, meeting suppose in K. The condition 
for this, if a, a' are the distances of the centres from a fixed point in the line of 
centres, and if the radii are c, c, is readily found to be 
(a _ a J = 2 (c 2 + c' 2 ). 
Suppose in general, that (given any two conics) the point P' is the intersection 
of the polars of P in regard to the two given conics respectively; then if P describes