and their Index Symbol. 135 



be investigated, <& m , 4>„ being given homogeneous functions in 

 x, y, z of the with and rath degrees, respectively. 

 The required solution is 



3> <£» 



u = 



+ 



+ u + n x + u r 



m{m-l){m-2) ' n(»-l)(n-2) 



7. Supposing two of the independent variables to vanish in 



the last example and one in each of the preceding, we are at 



once furnished with the solutions of the following ordinary linear 



differential equations : — 



dx* 



which are, respectively, 

 ax m 



z B -ri =ax m +bij n , 

 x* d ?y=ax m +by n , 

 x 2 — a —nx-~- + ny = 0, 



1 



dx 



y= 



y= 



m(m—l)(m — 2) 

 ax m bx n 



+ 



bx n 



+ 



»(n— l)(n-2) 



+ C + <V, 



+ C + C 1 a? + C^B*, 



m[m — 1) ' n(n—l) 



Now it must be remembered that the solutions given by the 

 symbol V are the same in form, no matter how large the num- 

 ber of independent variables may be. For instance, the solu- 

 tion of 



2 cPiu g dho ^dho_ 



Xl dx~? +X * dx£ + * 3 dx£ 



/ cPw d?w \ 



is exactly the same in form as that of 



■ = V m +V n . (a) 



namely, 



m[m— 1) n(n— 1) 



the only difference being in the number of independent variables 

 contained in u , u v 



Hence, in order to find (he form of the integral of an equation 

 of the class (a) containing any number n of independent vari- 

 ables, it is sufficient to have found the form of the integral of a 



