DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 81 



so that 



, , . 3H 



X = ~ </, P, tf ' > = ~ -- '' 



i , /B , 



~ -- * 2 x 2 '- 



Hence 



\P & + ? j X = 



n = 



But </> (or, what is the same thing in effect, G and H) must be such as to give 



dv dz 



~r n T ' 



du du 



so that 



n = '~~T~- > 



consequently we have 



I m n f" (u) 



i/ \ / 



p q I A 



which are the proper values of I, m, n, connected with v =f(u). 

 Similarly for the quantities a, b, c,f, g, It. 

 And if the value of H be required, it is determined by the equation 



33H - = /M = /L(!i), 



dr, 3 A " 1 + 7/ ' 



the integration of which is immediate. 



Application to V~v -f- K 2 v = 0. 



50. The integrable combinations of the subsidiary system in the case of the 

 potential-equation seem fortuitously obtained. As one other illustration, added 

 chiefly to show that the subsidiary system can be used in the mode indicated in 39 

 to express all the variable quantities in terms of a single quantity, we consider the 

 equation V 2 -v -f- K-V = 0, which is 



a + b + c + K 2 v = 0, 

 in the notation of the present paper. 



VOL. CXCI. A. M 





