214 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



In the case of an internal point the above four integrals may 

 be made to depend on the first. Calling 



du 



i. 



o J(a* + u) (b* + u) (c 2 + u) 



d _ ^ r _ i ____ dM 



8 (a 2 )" Jo 2 (a 2 + u) 7(a 2 + u) (6 2 + u) (c 2 + u) ' 

 and accordingly, 



9 (a 2 ) 



The integral <3> is an elliptic integral independent of a?, y, z, 

 and so are its derivatives with respect to a 2 , 6 2 , c 2 . Calling these 



L M N 

 respectively -r , -7- , -r , we nave 



a symmetrical function of the second order, and since L, M, N 

 are of the same sign, the equipotential surfaces are ellipsoids, 

 similar to each other. Their relation to the given ellipsoid is how- 

 ever transcendental, their semi-axes being 



7 



IV IV 



9$ Vo ** ' V 2 1* ' 



~V) a(6 2 ) 8(c 2 ) 



We have for the force 



37 ,, dV 



^ = My, - 5- = Nz. 



dy dz 



Hence, since for two points on the same radius-vector, 



#2 2/2 ^2 r 2 X* F 2 Z 2 T 2 



- = ^* = - J = - we have ^- 2 = -/=-=; = . 

 #1 yj *i n J?! F! ^ n 



The forces are parallel and proportional to the distance from 

 the center. 



114. Verification by Differentiation. For an outside 

 point, we have 



6 







\_ 



u c 2 + u V(a 2 + u) (6 2 + u) (c 2 + u) 



