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)},