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



its vertex is 



dm 



we have for an element of the homoeoid the attraction 



d<o(BP-DP), 

 in one direction, and 



dm (AP - CP) 



in the other, or in the direction PB, 



da>(BD-AC). 



Draw a plane through ABO, and let ON be the chord of the 

 elliptical section conjugate to AB. Since the ellipsoids are similar 

 and similarly placed, the same diameter is conjugate to the chord 

 CD in both. But CD and AB being bisected in the same point, 



AC = BD, 



and the attraction of every part is counterbalanced by that of the 

 opposite part. 



108. Condition for Infinite Family of Equipotentials. 



Although the equipotentials of an ellipsoid are not in general 

 ellipsoids, we may inquire whether there is any distribution of 

 mass that will have ellipsoids as equipotential surfaces. 



Let us examine, in general, whether any singly infinite system 

 of surfaces 



F(at!,y t z,q) = 



can be equipotential surfaces. If so, for any particular value of 

 the parameter q, V must be constant, in other words Vf(q). If 

 x, y, z are given, q is found from F (x, y, z, q) = and from that V 

 from the preceding equation. 



Now in free space, V satisfies the equation AF=0. But, 

 since F is a function of q only, 



dx dq dx ' 



d*V 



~* 



da? dx 



