202 ON THE DETEKMINATION OF THE ATTRACTIONS 



where, in virtue of (17) K must necessarily be a function of h 

 only ; and as the required value of <, if it exist, must be inde- 

 pendent of h, we have, by making h = in the equation imme- 

 diately preceding, 



k being the value #, and v'<A that of v</> when h = 0. 



We shall demonstrate almost immediately that every function 

 $ of the form (20), No. 9, which satisfies the equation (19), and 

 which therefore is independent of h, will likewise satisfy the 

 equation (18) ; and the corresponding value of K obtained from 

 the latter being substituted in the ordinary differential equation 

 (17), we shall only have to integrate this last in order to have a 

 proper value of V. 



9. To satisfy the equation (19) let us assume 



", Is 8 ,-?. 2 ) &, &, & ............. (20) ; 



F being the characteristic of a rational and entire function of 

 the degree 2y', and the most general of its kind, and f- p , f a , &c. 

 designating the variables in <f> which are affected with odd expo- 

 nents only ; so that if their number be v we shall have 



the remaining variables having none but even exponents. Then 

 it is easy to perceive, that after substitution the second member 

 of the equation (19) will be precisely of the same form as the 

 assumed value of <, and by equating separately to zero the co- 

 efficients of the various powers and products of , ? 2 ,...f g , we 

 shall obtain just the same number of linear algebraic equations 

 as there are coefficients in </>, and consequently be enabled to de- 

 termine the ratios of these coefficients together with the constant 

 quantity k . 



In fact, by writing the foregoing value of < under the form 

 * = SA mi , ,.... .&.-. ...... fc* ............ (20'); 



and proceeding as above described, the coefficient of 

 ?,""&"* ...... fc* 



