50 PROFESSOR A. R, FORSYTE ON THE INTEGRATION OF 



The system of equations now is 



1=0, J = 



dm dn _ dl dn 



dx " dy ' ~ dy " dx 

 do dv 



involving the quantities I, in, n, v, z, as functions of x and y. In all the derivatives 

 here contained, u is parametric ; and consequently all the constants that arise in the 

 integration are constants on this supposition : in other words, all of them are functions 

 of u. Consequently, when the integrals of the system are obtained, one constant (at 

 choice) can be taken to be u ; all the other constants are then functions of u, 

 arbitrary so far as the system is concerned ; and any arbitrary function of x and y 

 that occurs is also (possibly) a function of u. In order to determine the limitations 

 on the arbitrary functions, the equation 



dv dz 



= n 



du du 



must also be satisfied ; this equation will usually give relations among the arbitrary 

 functional forms, or will determine one of them. 



29. The relations thus obtained constitute an integral of the equation. For 

 suppose that in the expression for v we consider u eliminated in favour of z ; then 



7 OP , Be 7 Bv /' , 7 dz 7 \ 



dv = -5- occ + -5- cty + -5- aoj + o ay + -j" du . 



dt; By J Bx \f ' dii, J 



But also 



; 

 dv = 



whence 



do 



and therefore, comparing these with the equations of the system leading to the 

 integral form, it follows that 



ov Bo 



m = 3- > n = - - 



cy dz 



Next, take a quantity c such that 



dn dz 



du ~~ ~du 



