.Mr. Jolin Herapath on Functional Equations. 103 



member of the equation ought of itself to eliminate all the 

 quantities x, '^la.x, ■\i a^x, . . . except \{/ x ; and should become 

 simply \{/ X. If this does not take place, the equation is im- 

 possible. For that the right-hand result in a possible equa- 

 tion should reduce itself to ^x is evident; because to \t/a: 

 alone it is equal, and the functions of x with which rj/a^r, 

 ^x^x, . . . are in this final result incorporated,. contain every 

 possible change, by the substitution of u x, a^ x, . , . for jt, 

 which they can have, and are combined in every possible way 

 the properties of y admit : they want therefore no one ele- 

 ment or point in variety or form of combination which the 

 elimination of j:", 4/ « ^, vj/ «° .r, . . . that must unavoidably take 

 place requires ; and hence of course it ensues. 



Thus in ■i/ x •=^fx.'i^ax where o? x =■ x the condition of 

 possibility isy^r .fa. x ■=\; and if this does not hold, the equa- 

 tion is impossible. For instance, ya; = 2 giving ^I/ .r = 2v|/ « j; 

 is an impossible condition. So in the same equation, where 

 a" 0'=^, the condition of possibility isfx.fetx . . . .fa. ^~ x= 1> 

 without which it is impossible. And thus we may examine 

 the conditions of possibility in more complex cases. 



Of the Limitations to the Form of f iii (2). 

 In any equation of the form of 4* x =^4' «-^ ^ where u^ x=x 



n 



it is easily shown thaty '^ x = x, or that y must be a periodic 

 of the — th order. If therefore in (2) we regard the right- 

 hand member as a function of any term only, ■];a.^ x for ex- 

 ample, the other terms and the functions of x with which this 

 term is combined, must be so involved as with their respective 



properties to make it a function of the — th order. 

 This is evident in the equation 



where o^ x ^= x scnAfx is any symmetrical function of jr, a x, 

 a^x; and likewise in the equation 



•^ X =: fx."^ ax -T fx 

 where a.^ x = x, and the conditions of possibility arefx.fax 

 = 1, and fx.f ax = —fx. 



On the Number of Arbitrary Functions in the complete Solu- 

 tion of [2). 



For x in (2) put \(/~ x, and the equation becomes 



X =ry|\J/ .r, \I/«4/~ .r, . . . -^f OL^ ^~ x\ 



But 



