14 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 

 and we have the relations of identity, viz. , 



dh da dg dg 



dx dy dy dx 



db dh df df 



dx dy * dy dx 



df dg dc dc_ 



dx dy dy dx 

 Take first the form 



showing that z is any function of x + ,'/ The three deduced equations then become 



so that h g, b f,c f, are functions of x -\- y. 

 Hence, as, by the original equation, we must have 



we take, as integrals of the differential equations of the present type, 



h - g = F t (x + y, z), 



where F 1 and F 2 are arbitrary. Hence also 



~- (m n) = h g = 



(fYl Ttf\ b f 



oy 



a , 



y, z), 



F x (a; + y, 2), 

 Fj (x + y, z), 



It therefore follows that 



