ELIMINATION OF FUNCTIONS. 275 



256. "We will give one case in which more than three 

 variables are involved. Suppose 



F{u,x,y,z,$(a,&}} = Q .................. (1), 



in which <j> (a, /3) designates an arbitrary function of the two 

 quantities a and /?, which are themselves both known func- 

 tions of u, x, y, and z. If we differentiate (1) with respect 

 to each of the independent variables x, y, z, we obtain three 

 equations 



*?-0 ^-0 ^-0 (21 



& ' dy ~' dz~ ............... (2) ' 



In these equations, besides the arbitrary function <f>, we 



have its two derived functions -2- and -^ . Hence, between 



da. dft 



the four equations (1) and (2), we shall be able to eliminate 

 the three arbitrary functions, and arrive at an equation in- 



du du , du 

 ,*,y,z, , , and . 



EXAMPLES. 



1. Eliminate the constant from 



xy-c=(x + y] (c-1). 



Result. ( 2 + a;+l)+/ + + 1=0. 



2. Eliminate e* and cos x from 



y & cos x = 0. 



3. If of* 2ay a* 5 = 0, shew that 



x ^_dy = Q 

 da? dx 



4. If y = ae* sin nx, shew that 



T2 



