36 Ellipsoidal Configurations of Equilibrium [OH. m 



We shall write for brevity 



a z + \ = A, 6 2 + X = 5, c 2 + X=CM 



, j- ............... (53). 



[(a 2 + X) (tf + X) (c 2 + X)J* = (ABC)* = A J 



We shall take abc = r 3 . The matter of which the ellipsoid is formed 

 will be supposed to be homogeneous and of density /o, so that the mass M 

 will be given by 



M = 7rpabc = f 7J7>r 3 . 



The potential F of this mass at any external point as, y, z, is, by a well- 

 known formula, 



2 \ 



............... (54) 



in which the lower limit of integration X is the root of equation (52), and 

 so is the parameter of the confocal ellipsoid on which the point as, y, z lies. 



The potential F< of the mass at an internal point as, y, z, is 



and so is a quadratic function of x, y, z. 



35. To simplify the printing of integrals of the type just written down, 

 we shall introduce an abbreviated notation. Let us write 



(56) 



o A 

 and put further 



so that, for instance, equation (55) assumes the form 



Vi = - Trpabc (x 2 J A + y*J s + z*J c - J) ............... (57). 



It is easily verified that 



(58) 



or this can be seen from the circumstance that VFi must be equal to - 4>7rp. 

 We may also note the formulae 



(59), 



(60), 



l)J^+i = ^_ ............... (61) 



all of which are easily verified by algebraic transformations. 



With these preliminaries, we proceed to the three problems already 

 specified. 



