16 PROFESSOR A. B. FORSYTE ON THE INTEGRATION OF 



where G! and G 2 are arbitrary. But 



a- (m 1) = l> a = G 2 (x, y + z), 

 ox 



r* 



y- (TO l) = b h= Gj (x, y -f- z), 

 ^-( m -f) = f-g= G, (x, y -1- z), 



P*,, \ /.' V? I \ * J 



so that we have 



3,r ?y 3.t 



consequently there exists a function, say B (x, y + z), such that 



8B d 



U 2 = + 



and we then have 



, 



(jr., = -t- ri -^ 



da; r)// 



/ m B (x, y + s). 



We may proceed from this equation to the primitive. 

 As two distinct intermediary integrals, viz.: 



/ m = B (x, y + 2), 

 w, )( = <!> (.r -(- ?/, 2), 



have been obtained, it is worth noticing that they can be treated simultaneously, for 

 they verify identically the JACOBI-POISSON condition of coexistence. If we introduce 

 two new functions, and $, defined by the equations 



30 M 



9 (*)= - 



so that and < are arbitrary functions, then the simultaneous integral is easily 

 obtained, say by MAYER'S method,* in the form 



v = 6 (x +y,z) + <t>(x,y + z), 



which is the general primitive. 



10. "We next proceed to an example in which the given equation does not possess 

 an intermediary integral. It is not difficult to construct differential equations 



* Math. Ann., vol. 5 (1872), pp. 460-466: see also my ' Theory of Differential Equations,' Part I., 

 41-43. 



