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NOTE ON THE CARTESIAN WITH TWO IMAGINARY AYTAT.
FOCI.
[From the Proceedings of the London Mathematical Society, vol. in. (1869—1871),
pp. 181, 182. Read June 9, 1870.]
Let A, A', B, B' be a pair of points and antipoints ; viz.,
(A, A') the two imaginary points, coordinates (± ¡3i, 0),
(B, B') the two real points, coordinates (0, + /3) ;
and write p, p, a, a' for the distances of a point (x, y) from the four points respectively;
say
p = V (x + fiif + y\ er = V x 1 + (y + ST,
p = f(x — ftif + y\ a = V x? + (y — ST-
We have
p 2 + p' 2 = 2x 2 + 2 y 2 — 2/3' 2 = a 2 + a 2 — 4 S 2 ,
pp = + Si + yi) + Si — yi) ( x ~ Si + yi) ( x ~ Si - yi) = vg' ;
and thence
(p +p') 2 = (o' + a') 2 - 4S 2 ,
(p -p'T = (o- - o-'T ~ 4/3 2 ;
or say
p + p = V(cr + a') 2 — 4/3 2 ,
i (p - p) = V4S 2 -(<r — a') 2 •
The equation of a Cartesian having the two imaginary axial foci A, A' is
(p + qi) p + (p~ qi) p + 2k 2 = 0-
C. VII.
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