556 
[552 
552. 
ON A DIFFERENTIAL FORMULA CONNECTED WITH THE 
THEORY OF CONFOCAL CONICS. 
[From the Messenger of Mathematics, vol. II. (1873), pp. 157, 158.] 
The following transformations present themselves in connexion with the theory of 
confocal conics. 
The coordinates x, y of a point are considered as functions of the parameters 
h, k where 
x 2 y 2 
a + h h + h ’ 
x 2 2/ 2 
^+k + b + k~ ’ 
and then assuming ij =x + iy, y = x — iy (i = f(- 1) as usual), and writing c = a — h, 
we find 
h = £ (- a - b + (fa) + \ V{(£ 2 - c) (rj 2 - c)}, 
Jc = i (- a - b + %rj) - \ V|(£ 2 - c) (y 2 - c)}, 
whence if 
H = (a + h) (b + h), K=(a + k) (b + k), 
we have 
#=i if VW-«) + W(f-<=)!*, 
or, say 
\'<H) = 11!; \/<V -c) + n V(r - c)J, 
\W = i(fVW-<0-W(f a -c)},