OF ELLIPSOIDS 01' VARIABLE DENSITIES. 203 



will give the general equation 





, , 



Wto * * ' \ / 



**\ '2 -"wi, ma, ... mr+2, . . . m> 



a r 



the double finite integral comprising all the values of r and r', 

 except those in which r = r', and consequently containing when 

 completely expanded s (s 1) terms. 



For the terms of the highest degree 7 and of which the 

 number is 



7+1.7+2. ........ 7+*-l_7 



1.2.3 



the last line of the expression (21) evidently vanishes, and thus 

 we obtain ^7" distinct linear equations between the coefficients of 

 the degree 7 in < and & . 



Moreover, from the form of these equations it is evident that 

 we may obtain by elimination one equation in k of the degree 

 N 9 of which each of the N roots will give a distinct value of the 

 function </>M, having one arbitrary constant for factor; the homo- 

 geneous function </>M being composed of all the terms of the 

 highest degree, 7 in <f. But the coefficients of $M and Jc being 

 known, we may thence easily deduce all the remaining coeffi- 

 cients in <, by means of the formula (21). 



Now, since the N linear equations have no terms except 

 those of which the coefficients of <M are factors, it follows that 

 if k were taken at will, the resulting values of all these coeffi- 

 cients would be equal to zero. If however we obtain the values 

 of N 1 of the coefficients in terms of the remaining one A from 

 N 1 of the equations, by the ordinary formulas, and substitute 

 these in the remaining equation, we shall get a result of the 



form 



K.A = 0, 



where K is a function of & of the degree N. We shall thus 



