24 PROFESSOR A. R. FORSYTE ON THK INTEGRATION OF 



so that 



\&c 9z / ' ]1 



where / is an arbitrary function of x + y, Hence 



3* 92 



where G is arbitrary so far as this equation is concerned. 



Similarly, introducing a new arbitrary function g, defined by 



the equation 



a - 2k + b = 



leads to the equation 



a - 2ft + I - ( *- - j- 

 \cte oy 



Hence 



r/, 2 ) + ^ - <j, + F (x + y, z), 



where F is arbitrary so far as this equation is concerned. 



In order that the two equations, giving the values of I n and I m respectively, 

 may coexist, they must satisfy the PoissoN-JAOOBi condition (U, V) = 0, which, 

 when developed, gives 



/ 1 -/ 2 +G + S'i-5' 8 + F = 0; 



so that, taking account of the arbitrary character of the functions, we have 



F=-C/i-/l). G a -(ft -ft), 



Thus 



I - n = (y + z) (f n - 2/ 13 

 l-m = (y + z) (g u - 2g n 



It is easy to verify, not merely that these equations coexist, but also that each of 

 them satisfies the differential equation ; but neither is an intermediary integral in 

 the customary sense, for each of them includes two arbitrary functions of two 

 arguments. 



The equations are of the first order ; it is easy to obtain the primitive in the form 



v = 2/+ 2<j + (y + z)(f l -f, + (Jl - ft), 



where/j =/(# -f y, z) and g = y (x + z, y) are arbitrary functions, and f lt f z , y L , y.,, 

 are their respective first derivatives. 



