INVERSE FUNCTION. 35 



In equation (2) we may treat y as the independent variable 

 and x as the dependent variable, and we find, by Art. 47, 



From (2) x - 1 = + yl, 



therefore - = + /*, 



x-l 



Hence, from (4), = 



Comparing (5) with (3), we see that 



^ x = 1. 

 f&e dy 



The theorem which holds in this simple case we shall now 

 prove to be universally true. 



m dy dx 



60. To prove -/- x -j- = 1. 

 dx dy 



(1), 



since from this it follows that x must be some function of y, 

 suppose x = ty(y) ......................... (2). 



suppose x = y ......................... . 



Let x in (1) be changed into x + A#, in consequence of which 

 y becomes y + Ay, then 



y + Ay = (j> (x + Aaj) .......... . ........ (3). 



Now in (2) it may happen that x has more than one value 

 for any assigned value of y, but if the value of y in (2) be 

 the same as that in (1), then among the values which x can 

 have, one must be the value we supposed assigned to x in (1). 

 Hence we may suppose x and y in (2) to have the same 

 values as the same symbols respectively had in (1). In. equa- 

 tion (2) change y into y + Ay, where y has the same value 

 as in (1) and (3), and Ay the same value as in (3). Then 

 among the values which the dependent variable is suscepti- 

 ble of in (2)_, one must be x + A#, the symbols having the same 

 values as in (3). 



D2 



