156, 157, 158] POTENTIAL WITHIN ELLIPSOID. 419 



38) ' du 



o 

 and accordingly 



39) V = 



The integral is an elliptic integral independent of x, y, z, and 



so are its derivatives with respect to a 2 , & 2 , c 2 . Calling these respec- 



,. , L M N , 



tively 9 ) t we have 

 ^444 



40) V=itabc 



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 however trans- 

 cendental, their semi- axes being 



V* g ^ * 

 c<& 

 o 

 ^(tt' 2 ) 



We have for the force, 



3V 



V T 



42) <y-^==X=>yLx, 



Therefore, since for two points on the same radius vector, 



-17- - T7 - - ^ - 



X, F, Z t r t 



The forces are parallel and proportional to the distance from the 

 center, though not directed toward the center. 



158. Verification by Differentiation. For an outside point 

 we have, differentiating 34) 



a 



27' 



