395] 
65 
395. 
INVESTIGATIONS IN CONNEXION WITH CASEY’S EQUATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), 
pp. 334—341.] 
In a paper read April 9, 1866, and recently published in the Proceedings of the 
Royal Irish Academy, Mr Casey has given in a very elegant form the equation of a 
pair of circles touching each of three given circles, viz. if U — 0, V = 0, W = 0 be 
the equations of the three given circles respectively, and if considering the common 
tangents of (F=0, W = 0), of (1F = 0, £7=0), and of (£7=0, V=0) respectively, these 
common tangents being such that the centres of similitude through which they 
respectively pass lie in a line (viz. the tangents are all three direct, or one is direct 
and the other two are inverse), then if f g, h are the lengths of the tangents in 
question, the equation 
V (fU) + V (gV) + V (hW) = 0, 
belongs to a pair of circles, each of them touching the three given circles. (There 
are, it is clear, four combinations of tangents, and the theorem gives therefore the 
equations of four pairs of circles, that is of the eight circles which touch the three 
given circles.) 
Generally, if £7=0, V— 0, W = 0 are the equations of any three curves of the same 
order n, and if f g, h are arbitrary coefficients, then the equation 
V (fU) + f 07^) + V (h W) = 0, 
is that of a curve of the order 2n, touching each of the curves £7=0, V=0, W=0, 
n 2 times, viz. it touches 
£7 = 0, at its n 2 intersections with gV — hW=0, 
V=0 „ „ hW-fU = 0, 
W = 0 „ „ fU-gV= 0. 
c, VI. 
9