52 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



J + cl = 0, 



and eliminate the derivatives of I, m, n, with regard to x and y, we have the original 



equation 



F = 0, 



or the original equation is satisfied in virtue of the integral system. But. from this 

 integral system, the value of v is such that 





and therefore v is an integral of the partial differential equation. We consequently 

 have the theorem 



When an equation F = of the second order is transformed into J + cl = by 

 means of the equations 



dl 



dm dn 



p 



ax * ay 



dl dn 



and when, in the integral equivalent of the simultaneous system 



I = 0, J = "| 



dm dn dl 'dn 



dx " dy dy dx 



dv dv 



= I + np, = m + no 



dx dy 



all the arbitrary constants are made functions of a parameter u, and the arbitrary 

 functions of x and y are also made functions of u, subject to the equation 



dv dz 



= n , 

 du du 



(which, in fact, will generally determine either an arbitrary function or relations 

 among the arbitrary functions), then the value of v thus obtained is an integral of the 

 original equation F = 0. 



30. The integration of the equation F = is thus made to depend upon the 

 integration of a simultaneous system involving fewer independent variables, and 



