488] 
NOTE ON A RELATION BETWEEN TWO CIRCLES. 
13 
a line, the locus of P' is a conic passing through the three conjugate points of the 
given conics; if, however, the line which is the locus of P pass through one of the 
conjugate points, then the conic the locus of P' breaks up into a pair of lines, one of 
them a fixed line through the other two conjugate points, the other of them a line 
through the first-mentioned conjugate point. That is, if the locus of P be a line 
through a conjugate point, the locus of P' is a line through the same conjugate 
point; but in every other case the locus of P' is a conic. 
Reverting to the figure of the two circles, in order that it may be possible that 
the two lines AD and BG may be loci of points P, P', related as above, it is necessary 
that K shall be a conjugate point of the two circles; that is, if the two circles inter 
sect in points A, A' lying symmetrically in the radical axis, which meets, suppose, the 
line of centres in M, then it is necessary that K shall be one of the anti-points of 
A, A'; or, what is the same thing, the distance KM must be = i into MA or MA'; 
this condition, if as above (a — a') 2 = 2 (c 2 + c' 2 ), implies c 2 = c' 2 , and we have then 
(a — a') 2 = 4c 2 , that is, the circles must be equal, and the distance of the centres must 
be twice the radius, or, what is the same thing, the circles must be equal circles 
touching each other; when this is so, the two lines AD, BG (being then lines at right 
angles to each other intersecting in the point of contact), have, in fact, the above- 
mentioned relation. And it thus appears that given two circles, the necessary and 
sufficient conditions for the coexistence of the properties mentioned in the theorem are 
that they shall be equal circles touching each other.