DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 

 pW-ql=f-> qw + pl=- 1 ^, 



or 



''V 



w = -P ;/;, + 



Substituting these values above, we find 



denoting an arbitrary function of a;, ?/, M. 



The second of the three subsidiary equations similarly gives 



77 



denoting an arbitrary function of cr, y, u ; and the third of the subsidiary equations 



20 = (p, <!, ~ P^~ ' ffi 



gives 



n = 



\l> denoting an arbitrary function. 



These three arbitrary functions are not independent of one another. Comparing 

 the two values of 2f, 2g, 2h, respectively which have been obtained, we see that some 

 function <f> of x, y, u, exists such that 



