ELIMINATION OF FUNCTIONS. 269 



Hence this last equation is true whatever be the form of 



the function <f> ; for example, if z = log ( - ) , or z = sin - , or 



fx\ m 

 z = ( - , in each case we have that equation subsisting. 



247. Suppose u = <f> (v), where u and v are known func- 

 tions of x t y, and z, but the form of <f> is not given. The 

 variables x and y are supposed independent. If we differen- 

 tiate both members of the equation with respect to x and y 

 successively, we have 



du du dz i'/\ (dv dv dz ] 

 dx + ~dz dc = $ W [tic + dz dx] ' 

 du dudz , . . (dv dv-dz 



Therefore, whatever be the form of <, 



/du du dz\ /dv dv dz\ _ /du du dz\ fdv dv dz 

 \dx dz dx) \dy dz dyj \dy dz dy) \dx dz dx 



In other words we have eliminated the arbitrary func- 

 tion <. 



248. Suppose 



two known functions of x, y, z, which enter into an equation. 



F {#, y, z, <p x (otj), ^faOl == ^ \^)' 



<j and <f> 3 being arbitrary functions. If we form the equations 



f = . f"> W- 



= 0, -,-,=' 



dx* dx dy dy* 



we introduce the unknown functions 



and these, with ^(a,), and <^ 2 ( 8 ), form six quantities to be 



