PR WALLACE ON A FUNCTIONAL EQUATION. 627 



pendent variables, is the mrp same function, whichever of the two be reckoned varia- 

 ble, the other continuing constant. That is, supposing w^ and x, to be independent 

 variables, and t/ any function oioc. + a;, represented hjf(a)^ + x,), then 



This property, which is sufficiently known, may be exemplified by a particular 

 case. Suppose ?/=(a;„ + «^)", then, making x^ variable, and x^ constant, 



and making x, variable, and x^ constant. 



dy 



=zn{x^ + x)' 



L— 1 



dx, 



9. Applying now this property to the functional equation (A) ; making x^ va- 

 riable, and X, constant, we have 



dx^ 

 ^^¥^A=^) = cr {x, + x) + cf {x^-x) . 

 Again, differentiating the same function, and makiug x^ constant, we have 



^^f(x^) = c/'ix^ + x,)-cf(x^-x,}, 

 dx, 



^^^/W=«/" {x^+^)+cr{x^-x) . 



Now the right hand side of the second differential equation being the same on 

 either hypothesis, we have 



dx'^ ^^'' dxf •'^°^' 



and, putting y^ for/(^„), and y, for/(a?,), 



d'^yo \^ _ d'^y, 1 

 «^< y^ ~ dxf y, 



The two sides of this equation are functions of the same form, the one of x^ and 

 the other of x^ , and, by hypothesis, these quantities are independent of each 

 other ; therefore, each must necessarily be equal to some constant quantity, which 

 is the same for both : so that we have 



d^y 1 



. „° • — = a constant ; 

 d^o 'Jo 



and, in general, denoting/(a?) by y, 



d^y \ , , 



-~s ■ —= a constant. 



dx? y 



