OF ELLIPSOIDS OF VARIABLE DENSITIES. 207 



11. The value of < and H being known, we may readily 

 find the corresponding values of V and p. For we have im- 

 mediately 



-" 



(86) , 



and as the function < is rational and entire, and the partial dif- 

 ferential of V relative to h is finite, it follows that all the partial 

 differentials of V are finite ; and consequently, by what precedes 

 (No. 7) the condition (9') is satisfied by the foregoing value of 

 V, as well as the equation (2) and condition = V". Hence the 

 equations (b) and (c) No. 6 will give, since 



dV -^ 1 * dV dV 



and h must be supposed equal to zero in these equations, 



_dV 



since where h = 0, a r = a/; and therefore 



/22 



If now we substitute for V its value (26), and recollect that 

 n s + l is always positive, we get 



since it is clear from the form of H Q that this quantity may 

 always be expanded in a series of the entire powers of h*. In 

 the preceding expression, (27), H^ indicates the value of H 

 when h = 0, and <' the corresponding value of < or that which 

 would be obtained by simply changing the unaccented letter 



, ? 2 , into the accented ones /, f 2 ', f/ deduced 



from 



fry) '=<'; < = <&'; *.'=.''. 



