36 DIFFERENTIAL COEFFICIENT 



Hence x + Aa; = i|r (y + Ay) (4). 



From (1) and (3) 



Ay <j> (x + Aa;)* < (x) 

 ' ^ ^ ^^ ^ it (y). 



From (2) and (4) 



= T (6). 



Ay Ay 



In (5) and (6) the same symbols have the same values, and 



Ay Aa; 



since m that case -r 2 - x -r = 1, we have 

 A# Ay 



<f> (x + Ax) <f> (x} ilr (y + Ay) ty (y) 

 Aa; Ay 



Now diminish A# and Ay without limit, and we have 



or, as it may be written, 



dy dec _ 



61. The demonstration given in the last Article may 

 appear laborious. In reviewing it, the student will perceive 

 that this arises from the necessity of proving that the x, y, 

 Ace, and Ay, which occur in (5), have the same numerical 

 values as the quantities denoted by the same symbols respec- 

 tively in (6). This point is sometimes assumed, and it is 



AT/ A& 

 considered sufficient to say " since j- - x -^- = 1 always, we 



fl 77 {1 Y 



have, by proceeding to the limit, -, - x -y- = 1," but it would 



appear necessary at least that the assumption should be 

 noticed. 



62. Suppose z = <j> (x), 



y=V r (4 



so that y is a function of z, and z a function of x. It follows 

 that if we substitute for z its value in ty (z), we make y an 



