322 Mr. Herapath on the Solution of V x =■ x. [Nov. 



Article II. 



On the Solution of 4> n x = x. By John Herapath, Esq. 



(To the Editors of the Annals of Philosophy .) 



GENTLEMEN, Cranford, Oct. 5, 1824. 



Since the publication of Mr. Babbage and Mr. Herschel's 

 beautiful researches on periodical functions, the extension of 

 the functional calculus is become a subject of considerable inte- 

 rest. Among the first and most useful parts of functions stands 

 the solution of 



+" x = x (1) 



Indeed this solution is the hinge on which all further inquiries 

 must naturally turn. Limitations, therefore, in this part, una- 

 voidably beget limitations in the higher operations, and thus 

 deprive the calculus of its chief excellence, unbounded genera- 

 lity. All the solutions of (1) I have yet met with are confined 

 to the evaluation of 4- x from positive integral values of n. In 

 the following pages I have sought the value ofiJ/.r generally 

 from the simple condition 4-" x — x, without assuming any rela- 

 tion between v and n, or any limitation to their values. This I 

 have effected, first by indirect methods, as it has usually been 

 done, and then by a direct process extremely simple and general. 

 A few observations suggested by the preceding solutions are 

 afterwards added, respecting the number of arbitrary functions 

 in the complete solution of (1), which it is hoped will settle that 

 important question. 



Lemma. — -Ift}/ ' x = « x, a x being any function of x ; then 



n 



a}/'' x = a p x and ^" x — oC x, whatever be the values of p, n, v. 



For since V x is supposed equal to a x for all values of x, 

 i|, v must be of the same form as a; and, therefore, any operation 

 on 4/ as a whole by whatever index denoted must be identical 

 with the same operation on «. That is a 1 ' — (i/J or ^" p x = «" x. 



Whence if p — -, ^ n x = a" x. 



n 



v> 



§ 1. Assuming a x = b x, we get 

 a* x = ¥ x, 



a 3 X == b 3 x, 



and generally V x = b n x by the preceding Lemma for every 

 value of?/. If, therefore, 



. 4," x == «" x and b" = 1 , 

 we have 



V 



fx b l r . x (2) 



