OF ELLIPSOIDS OP VAEIABLE DENSITIES. 205 



But since a is independent of f t , f 2 , ...... f 8 , this last equa- 



tion is evidently identical with (18), since the equation (18) 

 merely requires that K should be independent of , f 2 , ...... f 8 . 



Having thus proved that every function of the form (20) 

 which satisfies (19) will likewise satisfy (18), it will be more 

 simple to determine the remaining coefficients of <j> from those of 

 <(Y) by means of the last equation, than to employ the formula 

 (21) for that purpose. 



Jc 

 Making therefore h infinite in (18), and writing in the 



place of K, we get 



s (s 1) 

 where (22) comprises the ^ combinations which can be 



formed of the s indices taken in pairs. 



If now we substitute the value of </> before given (20'), and 

 recollect that for the terms of the highest degree we have 

 2w r = 7, we shall readily get 



. ms ...... (22), 



from which all the remaining coefficients in </> will readily be 

 deduced, when those of the part </>M are known. 



10. It now remains, as was before observed, to integrate 

 the ordinary differential equation (17) No. 8. But, by the known 

 theory of linear equations, the integration of (17) will always 

 become more simple when we have a particular value satisfying 

 it, and fortunately in the present case such a value may always 



be obtained from </> by simply changing f r into - Jf r /2 . In 



\/(2,a r ) 



fact, if we represent the value thus obtained by H Q we shall 

 have 



dh 



