108 Mr. John Herapath on Functional Equations. 



which coincides with the solution obtained by owv first method. 

 Mr. Babbage (in Phil. Trans, for 1817), has given 



for the solution containing two arbitrary functions (p, <p,. It 

 would hardly be worth while, in a thing so easy, to spend time 

 in showing that this solution and mine are identical in point 

 of generality; 1 shall therefore, as I have only exemplified in 

 jieriodics of the second order, produce the testimony of a pe- 

 riodic of an indefinite oi'der. 



Let i> X =:J\r.^ u a-, where u^x = x, and the condition of 

 possibility \sfx.fa. x . . .fai'~ x = 1. A particular solution 

 of this equation is 



rl/ X = 1 +/x +fx.fa.x + . . .fx . . ./«"" X 

 And if this be multiplied by ^o: the proposed equation will 

 give 4)x = /^ctx; whence the complete solution is 



■\,x = ji+/^+y^./«^'+« .. .\-\^x-\-<pux+ . . . <p«" X ^ 



or, ■^x = <p X -\-fx . (^ux -\- fx .fa x . <p x^x + . . . 



which agrees precisely with the solution deduced from our se- 

 cond method. 



Thus it is evident, so far as these testimonies go, that they 

 confirm the truth of our conclusions, whatever be the order of 

 periodicity ; and if it were not for the troublesome operations, 

 more complex proofs could easily be given. Hence we may, 

 1 presume, regard the difficulties respecting arbitrary func- 

 tions and complete solutions, which have obscured the func- 

 tional calculus of one variable with one periodic and one un- 

 known function, as cleared, except inasmuch as they rest on 

 the imperfections of conmion algebra. 



It seems from what we have said, that those which have been 

 termed general solutions are in fact complete ones. ^^' e should 

 hence be induced to conclude that the solutions of functional 

 equations are but of two kinds, instead of three ; namely, par- 

 ticular and complete. 



Errata in my Paper, Phil. Mag, Sept. 1824, p. 108. 



Line 7, for catli a. x read fx . ct x 

 1 2, for .1- — ct X read x — fx . a. x 



XIV. On 



