410 VIII. NEWTONIAN POTENTIAL FUNCTION. 



152. 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 depending upon a parameter g, 



F(x, y, e, q) = 



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

 parameter g, V must be constant, in other words V is a function 

 of q alone, say V=f(q). If x, y, z are given, q is found from 

 F(XJ y, 2, q) = and from that V from the preceding equation. 



v Now in free space, V satisfies the equation z/F=0. But, since 

 V is a function of q only, 



dV dV dq 



~~O - == ~~7 - "C> - ' 



ox dq ox 



(W_dVd*qdqj)_ (dV\ 

 dx* ~ dq dx* + dxdx(dq) 



In like manner 



g 2 F _ 



" ~" ~ * 



dq dz* "+" \dz) dq*' 

 dV 



_ dV 

 Accordingly 



Q\ ^2 



3 ) T# = - 



dV dq\*dq) 



dq 



Now since F is a function of q only, the expression on the 

 right must be a function of q only, say <p(q). Consequently, that 



F(x, y, z,q) = 



may represent a set of equipotential surfaces, the parameter q must 

 be such that the ratio of its second to the square of its first differ- 

 ential parameter is a function only of g, 



