80 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



which is equal to 



d d\( f p & d*\.-\ 



ete +9*)(*4t + ^ S*-*W*J 



, d , d\ / d* P P \ 



and therefore 



(II 



= Act = g - - 

 du y du 



in virtue of the equation 



<'-< ' /2 ^ / / ' ; , ' ? \ , A'-* '^* 



P s + </ ^ + 2 ^ ^ + '/ T,} = i ^ + 



Similarly it can be verified that 



din , i/z iln, dz 



T~ J ~r ' r = c ~, > 



nil (in ilti ait 



in virtue of this equation ; therefore it is the only condition to be imposed upon 

 4> iii order that all the equations may be satisfied. 



49. Solutions of the equation a + I -f c = have been given in 31-38 ; it 

 is not difficult to verify that they are included in the preceding general form. Thus, 

 taking the solution 



v=f(u) 



and assuming that < is, as in 48, the function through which the term V () in v 

 is absorbed, it is first necessary to determine (f> so that 



,_o ''"< I d-tf> d-^> i // \ 



P~ ~j~z i *PQ ~T~i r <r ^T = J/ (U). 



This equation is easily solved ; and we have 



.2 



$ i ~^f( u ) + Gr (u, rf) + a;H (u, -rj), 



where G and H are arbitrary functions, and 77 denotes p'x + q'y, as before. 

 We have 



ff- ^ 



^$_ _ ( ,, 1 3 2 G ^ 9 2 H^ . 



