DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 29 



in the equation derived from 0, the latter should become an identity ; that is, we 

 should have 



30 30 ,od 30 [M 30 ,.30 30 \ , 30 fdli 



~ = h # E^- + / 5 h c - + K~ (i 



on \oz y ol .am on / act \dx 



30 <fo 30 30 </</ 30 ffy 30 



satisfied independently of the values of derivatives of h, c, y, with regard to x. It 

 therefore follows that, if a function of the suggested type should exist, it must 



satisfy the system of equations 



30_ 



3r" ~ ' 



30 30 



+ *7 - ' 



30 30 30 



and 



30 30^ 30_ .. 30_ v 30 30 



, 30 30 ,, 30 



and the number of functionally independent solutions of these homogeneous 

 simultaneous equations is the number of integrable combinations of the subsidiary 

 system. 



This system of partial differential equations of the first order must lie rendered 

 complete by associating with it the JACOBi-PoissON conditions. This complete 

 system, obtained by the regular processes, is without difficulty proved to be 

 equivalent to 



"3/6 =r 3" al' 



, o$_ 3_0_ a0_ _ _l _ o6_ _ l_ 30 . _ /9^_ __ 30 



s 3i ~" ~ 3^1 ~~ "" 3/i "~ // + ~ 3/ ~ ,y + a 3# \3 '~ db 

 and 



30 30_ 2/ - M - n 30 2a - 3/t. + & +/-.'/ 30 _ 



a "8 ' "(y + s)" 3 S ' y + s ob ' 



the latter being the modification of the last of the four initial equations. 



The complete system thus contains nine equations : it involves thirteen variables, 

 viz., a, b, c,f, <j, h, I, m, n, v, x, y, z ; and consequently it possesses four functionally 



