410 Mb green, ON THE DETERMINATION OF THE 



We shall demonstrate almost immediately that every function ^ of 

 the form (20), No. 9, which satisfies the equation (19), and which there- 

 fore 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 



<^ = ^(e.^ ?/, ?3^ ?/)?,.?„ &c (20); 



F being the characteristic of a rational and entire function of the 

 degree 2y', and the most general of its kind, and f,, ^„ &c. designating 

 the variables in which are affected with odd exponents only; so that 

 if their number be v we shall have 



7 = 27' + c, 



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 (p, and by equating separately to zero the coefficients of the various 



powers and products of ^1, |s, ^,, we shall obtain just the same 



number of linear algebraic equations as there are coefficients in <p, and 

 consequently be enabled to determine the ratios of these coeflScients 

 together with the constant quantity ^0. 



In fact, by writing the foregoing value of (p under the form 



</) = aS'^„„„, „„?.•"' ?."» ?»•" (20'); 



and proceeding as above described, the coefficient of ^ri ^/"t ^,'', 



will give the general equation 



K + 2)(m. + ]) 



^..i K+2)K + i) . 



"r 



