488 Lit egr ability of Partial Differential Equations^. [May 16, 



But F contains only ccyz ; hence, in order tliat this last equation may hold, 

 the coefficient of p must = 0, i. e. we must have 



0=±2aF(.^) + (^/(3 + a^^, (8) 



in which case (7) reduces to 



O=F'(«/) + aF'(^0-F.F'W-y^^ (9) 



F must satisfy both (8) and (9). 

 By integration of (8) we get 



Substituting this in (9), observing that 

 we get 



= F',,)T.F.(.)+||g + g, 



(10) 



But, since F^ contains x and y only, in order that this may hold we must 

 have the coefficient in it of = 0, i. e. 



dx 2\a dx) 2 Vcc dx ) 

 in which case (10) becomes 



= F',(y) + aF/(.)±^g + g.F.; 



whence by integration we get 



where ^ is arbitrary. 



Hence we get for the first integral of (1) ivhen (10) is satisfied^ 



whence by ordinary integration we obtain the complete integral of the form, 



