113] ELLIPSOIDAL COORDINATES. 337 



Accordingly we have the identity 



z\ F(o \ - x * , y" i = 



- - 1 r " ~ 



Multiplying this by p + a 2 and then putting p = a 2 we get 



2 _ (a a + a)(a a + fO(* a + 'Q 



r (a 2 -& 2 )( 2 -c 2 ) 

 and in like manner 



o 



* ' 



(&' -<>)(&-<) 



_ . 



If A, /i, v are contained in the intervals specified in 73, these will 

 all be positive, so that the point will be real. 



If we insert these values in d%, we shall have hi expressed in 

 terms of A, ^, v. 



This is more easily accomplished as follows. 



Differentiating the above identity 6) according to p, 



- ft) (g - y) f 1 1 



If we put p = A, all the terms on the right except the first, b.eing 

 multiplied by Q A, vanish, and we have 



, x* 



22 " 1 " 



The expression on the left is Therefore 



In a similar manner we find 



and the parameter of any function F(A, ^, v) is 



WEBSTER, Dynamics. 22 



