12 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



7. The first of the equations, A = 0, belonging specially to the postulated equation 

 V = 0, can be expressed in a different form. The quantity z is regarded as a function 

 of x, y, u ; and p, q, denote the values of dz/dx, dz/dy, respectively, when u is con- 

 sidered constant. Let the equation connecting z, x, y, u. be given (or taken) in the 

 form 



M = u (x, y, z) ; 



then we have 



CM 3 ou CM 



=r- + p K- = 0, 3- + Q a~ = 0. 



ex L 0j ay * cz 



Substituting for p and q, the equation A = becomes 



_L H a. TJ _u r a. v j. r 



+ rl 5- -. -- \- r> 3- -\- IT 3- -5- -f- r 5 2 T ^ I a 



ox oy \ dy / dx az oy dr \v 



which, after the preceding explanations in 4, is an equation satisfied by an argument 

 of an arbitrary function in the integral of the differential equation 



It is not difficult to prove that this equation is invariantive for all changes of the 

 independent variables. For suppose them changed according to the transformations 



x = (x, y, 2) , 

 y' rj (x, 11, z) , 

 z' = (x, y, z) ; 



and let ,, /,;,... denote derivatives of f , . . . while I', m', n', a' . . . denote deri- 

 vatives of v with regard to the new variables. Then 



I, m, n = ( &, >?.,, ^. % I', tn', n') ; 

 Vy> ^y fey 



and 



a = (<*',l>',c',S' i g',h r 

 h = (.', 6', C ', /', g,', /,' 



the terms represented by -f ... being terms involving the derivatives I', m, n', of 

 the first order only. If, then, the differential equation 



F (a, b, c,f, y, h, I, m, n, x, ij, z) = 

 becomes 



