336 VIII. NEWTONIAN POTENTIAL FUNCTION. 



A -i 



so that we may write for the direction cosines of the normal, 



COS 



yd i 



3) cos (my) = 



cos 



Now as we move along the normal, we have 



*&i 

 dx = dn cos (**#) = ^ 2 , ^ 



* 

 dy = 



^d; 



6?^ = dn cos (WA^) = 2 . 



C ~j~ 



Inserting these values in 2), 



x* , 2/ 2 



-|(^fiy 2 + (^ 



so that 



i 



2 " 



In order to express this result in terms of the elliptic coordinates 

 alone we may express x, y, 8, in terms of A, p, v. Observe that the 

 function 



BY N _ 3? y* , z* -i 



- -* + 2 + 2 - 



has as roots A, ^, v, and being reduced to the common denominator 



( e + O(e + & 2 )(9 + c 2 ) 



has a numerator of the third degree in Q. As this vanishes for 



9 = 1, 9 = P> 9 = v , 



it can only be 



- (? - ).) (Q - p) (Q - v). 



