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LVII. On the Transformation of the Solutions of a Periodical 

 Functional Equation. By John Herapath, Esq. 



T F ■]> x = x, I have found by a very simple and direct pro- 

 cess that the complete solution is 



.v I Ikvr , . -1 -1 > 



4/x = 4>sin<— 1- sin f x± (1) 



where tt is the semi-perimeter to radius 1, Jc any integer, and 

 f any arbitrary function; also v, n any numbers whatever, 

 rational, irrational, or imaginary. The function sin may also 

 be changed into cos, tan, sec, &c. if we please ; and the only 

 condition that appears necessary, if $ should contain an in- 

 verse circular function, is not to blend the operations, but to 

 let each act separately and distinctly. 

 Let us denote ( 1 ) by 



• v /■" — * 



V x - <Pf $ x (2) 



and in any other case, for instance, when ty t x = x, let it be 



, V V — \ 



%*~ <Pifi 4>/ x (3) 



Then if in (2) v be expounded by — we shall have 



n J" 



\J/r X = <Pf r <J> — *•* 



«_ in 



which by (1) is equal to fyx. Therefore 4>~ = ^ and \|/ r = 



>J/, , that is, a periodic function of any order is a given order of 

 a periodic function of any other order, both taken completely. 

 This result is also easily obtained a priori; but there is an- 

 other consequence flowing from it, that seems, to me at least, 



of a more novel character. For instance, because \J/ r s = ^ 



ff^'f x=J, ' x=<pf<p ?f,<p~ X x=$f<p- 1 <p l f- s 



<p, x: that is $ = ?,. In other words, if there be any two ar- 

 bitrary functions <p, <p t perfectly unlimited, then is $ = $,; or they 

 are identically the same. This singular consequence is evident, 

 if we consider that a function perfectly arbitrary must virtually 

 contain at every instant every form that can be given it. Two 

 such absolutely arbitrary functions must therefore simulta- 

 neously comprehend all and the same possible forms, and 

 consequently be equally and identically the same. I do not 

 mean to contend that an arbitrary function may not have at 

 any time any particular form the problem requires ; nor that 



two 



