ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 415 



r=i/^ = jir^//./-gj£2^ «6), 



and as the function (p is rational and entire, and the partial differen- 

 tial of f^ relative to h is finite, it follows that all the partial differ- 

 entials of F^ are finite; and consequently, by what precedes (No. 7.) 

 the condition (9') is satisfied by the foregoing value of F', as well as 

 the equation (2) and condition = F". Hence the equations {b) and 

 (c) No. 6 will give, since 



du- "V ^' ~^) Y" ^■d^~~dh\' 



and h must be supposed equal to zero in these equations 



- r f^^ii) 



p' = — , A. .---V^-^^ (where h = 0); 



since where A = 0, a, = «/ ; and therefore, 



1 - 2/^' ^^ = 1 - ^r' V = ^'. 



If now we substitute for V its value (26), and recollect that « — * + 1 is 

 always positive, we get 



-r(^) ^ 



27r^r 



(^4^) 



since it is clear from the form of Ho that this quantity may always 

 be expanded in a series of the entire powers of A^ In the preceding 

 expression, (27), H^ indicates the value of Ho when h = 0, and (p! 

 the corresponding value of or that which would be obtained by 



simply changing the unaccented letter fi, ^2, ^, into the accented 



ones ^1', f/, ?/ deduced from 



(7) x; = a,'?/ ; x.^ = «; ^/ ; x/ = «/ ^;. 



