Mr.- John Herapath o« Functional Equations. 107 



As a second example, let us take 



"^1 X + fx .■^ U.X =yj X 

 a.f being still a periodic of the second order, and the condi- 

 tions of possibility fx .fx x = 1, and fx .f u, x =y" .r. The 

 simplest solution of tliis equation is \fx. If therefore we 



seek the complete solution by assuming -i^ x =. ^- — f- f J", we 

 shall ultimately have 



(p r + fx .^ XX =. 

 an equation of precisely the same form as (8). Hence the com- 

 plete solution of our last equation is 



AfX = \fx-ir{\—fx).\^x + ^xx\ or = ^y;x + <px— /jc.faa: 



the latter form being that which I have deduced by our se- 

 cond method (Phil. Mag. for September 1824), and the former 

 being Mr. Herschel's (Spence's Essays, p. 163) under a de- 

 finite form. In comparing these two forms of solution part 

 by part, instead of the whole with the whole, I concluded the 

 latter was more geneml than the former, whereas they are pre- 

 cisely equal. 



Had we sought in the last example the complete solution 

 by assuming it = \fx.<^x, instead, of = \fx + ^ x^ we 

 should have found it 



r{/x= [i + (px— ($:«j:} .fx 



the simplest expression I have yet seen for it. 



The subsequent example I introduce more for the sake of 

 exhibiting another method of obtaining the complete from a 

 particular solution, than from any necessity of its testimony. 



Let , ^f n — \ \ 



^'^ =/l-^» ajr, . . . a x^ 



be a particular solution of any functional equation. Then, if 

 <f be perfectly arbitrary, and 



/ {4; J, 4^ ax, ... $a"~^x} 

 will satisfy the conditions of the proposed equation, it is the 

 complete solution. For since the conditions are satisfied, and 

 the expression contains an arbitrary function of every trans- 

 formation X can have, from x to « ~ .r, it nnist, by what we 

 have already demonstrated, possess all the properties of the 

 con)j)lete solution. Now a particular solution of 



rj/ .r'*+ ^^ a x" = 1 is ^J/x = I -^^^ J "^ 

 and iherelbre the complete solution is 



O 2 which 



