6 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



When p - q = 0, then z is a function of x + y. Also the second equation becomes 



rf / \ " i \ 



- (m n) = -~(m n), 

 dy^ A 



so that m n is a function of x + y. We infer, from the general considerations 



adduced, that we may take 



,n n = <& (x + y, z), 



where * is an arbitrary function. 



When q 4 1 = 0, then y + z is a function of x. Also the second equation is 



dm dm dn dn __ 



" ' 



which, by means of the identical condition, can be transformed to 



dl dm_ 

 -dy+dy- ' 



so that I in is a function of x, and the corresponding inference is that 



I - m = (x,y + z), 



where is an arbitrary function. 



Each of these is an intermediary integral, and the integration can be completed.* 

 3. Passing now to the generalisation of DARBOUX'S method, we change the variables 



from x, y, z, to x, y, u, where u is a function of x, y, z, as yet undetermined, so that 



also z is a function of x, y, u, not yet determined. For the consequent variations of 



2 when x, y, u, vary, write 



dz dz 



and when x, y, u, are the variables, denote the variations of the other quantities by d. 

 Thus 



The example is discussed (and the solution completed) in another connection in 9. 



