313] ATTRACTION OF ELLIPSOIDS. 581 



(6). 



The potential energy of the mass is given by 



..................... (7), 



where the integrations extend over the volume. Substituting 

 from (4) we find 



V= f 



(8). 



This expression is negative because the zero of reckoning 

 corresponds to a state of infinite diffusion of the mass. If we 

 adopt as zero of potential energy that of the mass when collected 

 into a sphere of radius a, = (abc) 3 , we must prefix to the right- 

 hand of (8) the term 



(9). 



If the ellipsoid be of revolution, the integrals reduce. If it 

 be of the planetary form we may put 



w v f^ 



and obtain* = /3 = (f 2 + 1) f cot" 1 f f 2 , 



(10), 



-%} (11) ' 



-'? (12), 



provided the zero of V correspond to the spherical form. 

 For an ovary ellipsoid we put 



^5-(zl?o (13), 



* Most simply by writing c 2 + X = (ti 2 -c 2 ) u\ 



