DR WALLACE ON A FUNCTIONAL EQUATION. 6*27 



pendent variables, is the ve7y same function, whichever of the two be reckoned varia- 

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

 variables, and y any function of x^ + x, represented by/(^, + x), then 



This property, which is sufftciently known, may be exemphfied by a particular 

 case. Suppose y^{x^ + x;f, then, making x^ variable, and x, constant, 



and making x^ variable, and x^ constant, 



ax, 

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

 riable, and X, constant, we have 



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



-^-^f{^o)^cf{x^ + x)-cf'{x^-x), 



dx 



dx? 



7W=C/"(^o + ^.)+c/"(^o-*.) . 



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

 either hypothesis, we have 



and, putting y^ for/(^J, and y, for/ (,27,), 



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



<?«' Vo 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 ; 

 dx\ y^ 



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



d^y \ , , 



—~ . —= a constant. 



dxr y 



