ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 411 



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 y + s — l _ ^ 



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

 obtain JV distinct linear equations between the coefficients of the degree 

 7 in <p and ko. 



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

 may obtain by elimination one equation in ko of the degree JV, of 

 which each of the iV roots will give a distinct value of the function 

 (p^'y\ having one arbitrary constant for factor; the homogeneous function 

 ^''1'' being composed of all the terms of the highest degree, 7 in (p. 

 But the coefficients of (p'-^'' and kg being known, we may thence easily 

 deduce all the remaining coefficients in (j>, by means of the formula (21). 



Now, since the A'' linear equations have no terms except those of 

 which the coefficients of ^'^^ are factors, it follows that if ^0 were taken 

 at will, the resulting values of all these coefficients 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 iV- 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 jRT is a function of ka of the degree iV. We shall thus have 

 only two cases to consider : First, that in which A = 0, and consequently 

 also all the other coefficients of 0*^' together with the remaining ones 

 in <p, as will be evident from the formulae (21). Hence, in this case 



= 0: 



Secondly, that in which kg is one of the iV roots of = K, as for 

 instance, ko in this case all the coefficients of will become multiples 

 of A, and we shall have 



Vol. V. Part III. S« 



