268 



ELIMINATION OF FUNCTIONS. 



245. Not only constants may be eliminated, but functions. 

 Suppose, for example, 



- y = sin x ; 



then j^ = cos x 



ax 



= V(i-2/ 2 ); 



therefore v a + I ~r ) 1 = 0. 



Vaay 



Hence the function sin x has been eliminated. 

 Again, let 



therefore -^- = f 1 -f tan 2 (x + y}}\ 1 + -r\ 



C1T \ ft "7* I 



Hence tan (x + y) has been eliminated. 



In these examples given functions have been eliminated : 

 we proceed to cases in which unknown functions are elimi- 

 nated. ' 



fx\ 

 246. Suppose z = <j> ( - ) , where <f> denotes some function 



the form of which is not given, and which is therefore called 

 an arbitrary function. The variables x and y are supposed 

 independent. 



Put -=*: then 



y 



therefore x ^- + y -r = 0. 



oic * ay 



