76 PROFESSOR A. R, FORSYTE ON THE INTEGRATION OF 



In the subsidiary equations and in the relations of identity, the derivatives with 

 regard to x and y are framed on the supposition that u is constant. Now the 

 complete solution of 



i O i i r\ 



p- + q~ + 1 = 

 is given, by CHARPIT'S process, in the form 



p = constant, q = constant, z px qy =. constant. 

 But these constants arise when u is constant ; hence we may take 



zpx qy- u, 

 where p = p (it.) and q = q (u) are arbitrary functions of u subject to the condition 



p" + f + 1=0. 



From the first of the three subsidiary equations, remembering that p and q are 

 now constant so far as concerns derivatives with regard to x and y, we have 



so that some function w of x, ?/, it, exists such that 



dw dw 



a pa = -- ' h no = -- - 



1 '' i/i/ IJ ill- 



Also we have 



,n <n 



r = _ , h + qg = _. , 





so that 



<lw dl 



2<( = T + Tx ' 



dw dl .1 

 f / y + ^ I 



' 



From the last we have 



dw dl dw dl 



that is, 



~ (pw - ql) + '^ (pi + qw) = 0, 



so that some function 6 of x, y, u, exists such that 



