272 ELIMINATION OF FUNCTIONS. 



This result involves only the same arbitrary functions as 

 the original equation, namely, 



it also involves -j- and -y- ; we may denote it by 



dz 



dy' ^' &^' ...... 



Differentiate this equation with respect to x and y as 

 before ; thus we obtain another result which involves only 

 the same arbitrary functions as the original equation. By 

 continuing the process until we introduce the differential 

 coefficients of z of the n th order, we find that we have on the 

 whole n+ 1 equations, from which the n arbitrary functions 

 may be eliminated. 



252. Suppose we have two equations of the form 

 F{x, y, z, a, &(a), <k(a), ...... <()} = 0, 



f{x, y, z, a, <k(a)> <, ...... ()} = 0, 



where a is an unknown function of x, y, and z, and fa , < 2 , ...< 

 denote arbitrary functions ; and let it be required to eliminate 

 a and the arbitrary functions of a. In this case also we shall 

 find that it will be sufficient to proceed to the derived equa- 

 tions of the n. th order inclusive. 



As in the preceding Article we differentiate the first equa- 

 tion and thus obtain 



dF dF dz da dxdz_ 



dx dz dx dx dz da . 



dF "d^ 



dy dz dy dy dz dy 



But as a is not a known function the right-hand member of 

 (1) is not a known function. But from the second of the 

 given equations we obtain in the same manner 



dot. dadz df df dz 



dx dz dx _ d - dz. dx , Q . 



da. da. d,z~ df df dz ..... " ( ~ )] 



dy dz dy dy dz dy 



