102 Mr. John Herapath on Functional Equations. 



" This is the converse of M. Arago's experiments, in which 

 he shows the effect of copper and other metallic rings, in di- 

 minishing the number of oscillations of a magnetic needle. 



" 14'. If, instead of a horse-shoe magnet, the contrary poles 

 of two bar magnets be used, the effect is the same as before ; 

 but, if the poles of the same name, viz. both north or both 

 south, be employed, then the effect is scarcely perceptible. 

 This is an important result, as it shows that the effect is not 

 due to any kind of resisting medium, as was supposed in the 

 first instance." — Editi. Phil. Journ. 



XIII. On the Conditions of Possibility, Arbitrary Functions, 

 and Complete Solutions of Periodical Ftmctional Equations. 

 By John Herapath, Esq. 

 COLUTIONS of functional equations have been considered 

 ^ to be of three kinds ; particular, general, and complete. It 

 has also been usual to consider the complete solution of any 

 equation of the first order, as for instance of 



Y {x,^x,^ctx,-\,oc'x, . . . ^ct'~^x]=0 (1) 



to contain n — \ arbitrary functions; and with respect to the 

 form of F, it seems to have been tacitly admitted to be quite 

 unlimited. Having been led by my inquiries to different con- 

 clusions, I purpose in the present paper to examine the con- 

 ditions of possibility of (1), the limitations to the form of F, 

 and to give a simple direct method of obtaining the complete 

 solution. I regret only that my very confined limits will oblige 

 me to be less explicit in my exemplifications than I could wish. 



Of the Conditions of Possibility. 

 When any equation of the form of (1) is given, it may be 

 put under the form of 



■\ix ■=f\x,'^a.x,'^ o^^x, . . . \I/a"~ or] (2) 



and there must simultaneously subsist the ?^— 1 following 

 equations, 



•^ax =if\a.x, "^ o^x, "^ ct^ X, ... \(/^} 



■^ cc X = f \(X^x, 4/ a^ jr, rj/ a* X, . . . \l/ a Jr] 



(3) 



^oJ'-'^x=f{oi.''-\,^>x,-]>ux,. . . ^>ol'^-'^x\ 



If now these 71 — 1 right-hand functions be substituted for 

 their values '^ax,'^»^x, . . . in (2), the resulting right-hand 



member 



