485] PROBLEMS AND SOLUTIONS. 567 
corresponding to an integral equation 
F (i|r) — F(y\r') = F (jx), 
the modulus of the elliptic functions being 
_ V8 ac 
’ " V{(2c + af-b 2 ]’ 
In the problem above considered the modulus is 
_V{ 4c 2 -(a + 5) 2 } 
’ V{4c 2 -(a- b) 2 } ’ 
and it is not very easy to see the connexion between the two results. 
[Vol. vi. p. 81.] 
Theorem: by Professor Cayley. 
If (A, A'), (B, B') are four points (two real and the other two imaginary) related 
to each other as foci and antifoci (that is, if the lines A A', BB' intersect at right 
angles in a point 0 in such wise that OA = OA' = i.OB = i.OB'), then the product 
of the distances of any point P from the points A, A' is equal to the product of 
the distances of the same point P from the points B, B'. 
In fact, the coordinates of A, A' may be taken to be (a, 0), (— a, 0), and those 
of B, B' to be (0, at), (0, — at); whence, if (x, y) are the coordinates of P, we have 
(AP ) 2 = (x — a) 2 + y 2 = (x — a + iy) (x — a — iy), 
(■A'P) 2 = (x + a) 2 + y 2 = (x + a + iy) (x + a — iy), 
{BP ) 2 = x 2 + (y — ia) 2 = (x + iy + a) (x — iy — a), 
(B'P ) 2 = x 2 + (y + ia) 2 = (x + iy — a) (x — iy + a), 
from which the theorem is at once seen to be true. 
An important application of the theorem consists in the means which it affords 
of passing from the foci {A, B, C, D) of a bicircular quartic, to the antifoci (A, B) and 
(C, D); viz. if these are {A', B\ G', D'), then the equation l\J{A) + m\/{B) + n *J{C) = 0 
must be transformable into l'\j{A’) + m'\f(B') + n' \Z{C) = 0. Writing these respectively 
under the forms 
PA + m 2 B - n 2 C + 2bn V(AB) = 0, l' 2 A' + m 2 B' - n' 2 C' + 2I'm' *J(A’ff) = 0, 
the two radicals \J{AB), \f(A'B') are identical; and the remaining terms in the two 
equations respectively are rational functions, which when the ratios V : m' : n' are 
properly determined will be to each other in the ratio Im : I'm'; the two equations 
being thus identical.