OF ELLIPSOIDS OF VARIAPLE DENSITIES. 201 



The last equation may be put under the abridged form, 



provided we have generally 



^ . f. d V . Tr I/-, fz K? <+l a r f 2 , ^r - 2 > \ 



coefficient of -^ m y ^= U - 6- - ^ f / 4- ^ f r , 

 ay tt r \ a^ a r j 



. . , 



coefficient of ,. . in F= 



a r a? 



^ . ,. dV . TT r f t <* a / Z a r'*\ 



coefficient of ^r in y V -^ n + 2, -- ^ 

 dl; r a? \ a/ a*J 



3loreover, when we employ the new variables 



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



where the values of the variables f x , f 2 ,...f s must be such as 

 satisfy the equation v 2 = 0, whatever h may be ; and as n s + 1 

 is positive, it is clear that this condition will always be satisfied, 

 provided the partial differentials of V relative to the new vari- 

 ables are all finite. 



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

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



08); 



H depending on the variable h only, and </> being a rational and 

 entire function of f 1? f 2 ,...f of the degree 7, and quite inde- 

 pendent of h. 



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



we readily get 



= v</>-*0 ........................ (18); 



