108, 109] ATTRACTION OF ELLIPSOIDS. 207 



which is independent of /JL and v, and hence the system of ellipsoids 

 X can represent a family of equipotential surfaces. We have 



(8) 



= log V(a 2 



(9) 7 = 4 [7= -- =+5. 



JVa 2 + 



The constant B must be such that for X = oo , which gives the 

 infinite sphere, F= f O. This is obtained by taking the definite 

 integral between X and oo , 



(10) " - r ds 



X being taken for the lower limit, so that A may be positive, making 

 V decrease as X increases. V is an elliptic integral in terms of X, 

 or X is an elliptic function of V. For 



dV A 



1) ^ ) = (a 2 + X) (6 2 + X) (c 2 + X), 



a differential equation which is satisfied by an elliptic function. 

 We may determine the constant A by the property that 



\i-rn (rV) = M, 



x= 



or that lim (V 2 -5 ] = - M cos (rx\ 



,.= \ 9# / 



We have 





(a 2 + X) (a 2 + X) (6 2 + X) (c 2 + X) 



, i 



OF THF ' y > 



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