128 NATURE OF FUNCTIONAL EQUATIONS. 



preceding example the previous operation is known ; but this is 

 not essential. Thus in 



+#=0 ........................... (4) 



the previous operation denoted by the right-hand <j> is unknown. 

 The operation -*- may enter into equations where the un- 



known operation is not performed on the base. Thus we may 

 have an equation of the form 



Equations (3), (4), (5) are functional equations; (3), (4) are 

 ordinary functional equations ; (5) is a differential functional 

 equation ; (3) is said to be of the first order, (4) of the second. 



The introduction of the functional notation appears to be 

 sometimes taken as the essence of functional equations ; but if 

 we wrote (1) and (2) thus, 



}<(a)} 2 + a<(a) + & = .................. (1)', 



3*(a)-a = .................. (2)', 



they would still be perfectly distinct from (3) or (4) or (5). 

 The name functional equation is not happy; it refers to the 

 notation, and not to the essence of the thing. 



A question now arises : To what class shall we refer equa- 

 tions in finite differences ? These are generally of the form 



*>,y,,y. )=<> .................... (6), 



where y x is an unknown function, say < (x) of x ; so that (6) 

 may be written thus, 



Here the unknown operation is <, which in the case of <f> (x + 1) 

 is performed, not upon the base x, but on x+1. Thus it 

 appears, that equations in finite differences are only a case of 

 ordinary functional equations of the first order : and this is the 

 reason why, in researches on functional equations, we per- 

 petually meet with cases in which they may be reduced to 

 equations in finite differences. 



