ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 409 



coefficient o£^-mvV=~ {1 -^^'-2.'*' 1^ ^ "' + ^ ^"}> 

 coefficient of ^, i» V ^ = - ^. lU 



coefficient of -j^ in vF=-^|-» + 2 ^ ^i. 



Moreover, when we employ the new variables 



du " y- ^ ^ . ; • Y' a? d^r dh ]' 



and therefore the condition (9) in like manner will become 



— -(>-^r"i^ff-^} «'»^ 



where the values of the variables ^1,^2, ?, must be such as satisfy 



the equation i;" = 0, whatever h may be; and as n-s-\-l is positive, it 

 is clear that this condition will always be satisfied, provided the partial 

 differentials of V relative to the new variables are all finite. 



8. Let us now try whether it is possible to satisfy the equation 

 (2'") by means of a function of the form 



r^Hct> (/?); 



H depending on the variable h only, and cp being a rational and entire 

 function of ^1, f^, f, of the degree 7, and quite independent of h. 



By substituting this value of V in (2'") and making 



^ d'H ( ^«:^ dH , „ ,,„^ 



we readily get 



= v<^ - '«P (18); 



where, in virtue of (17) k must necessarily be a function of h only; 

 and as the required value of (p, if it exist, must be independent of k, 

 we have, by making h = in the equation immediately preceding, 



= v'0 - ko(p (19); 



ko being the value k, and v'^ that of v^ when h = 0. 



