Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

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448. 
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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