412 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Al _ d L 1 -I/ (gt+l^ 



"Sir^i K Ql)i^-f)( 



1 I 1 1 1 \ _ m 



2 a'-V 2 - 2 ~ 



which is independent of /i and i>, and therefore the system of 

 ellipsoids A can represent a family of equipotential surfaces. We have 



9) w ji - + + 



10) V=AC- 



J V( 



J5 must be such a constant that when A = oo, which gives the infinite 

 sphere, F= 0. This is obtained by taking the definite integral 

 between I and oo. 



' 00 



11) V-A C- = ds 



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

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

 or A is an elliptic function of F. For 



J dl 



ia\ A*( d 



A \d 



a differential equation which is satisfied by an elliptic function. 

 We may determine the constant A by the property that 



lim(VF) = M, 



or that ~ v 



lim (r 2 -*rA = M cos (rx). 



r=<x> > i , . ^ 



We have 



^F ^ra^ 



[by 73, 86)] 



