246 Mr. Herapatk*s Repti/ to Mr. Horner. [April, 



17. Little needs to be said on the inferior sources of perplexity, 

 which have been imagined. It will have occurred to the reader, 

 even in Art. 1, that^' x having two distinct meanings may be 

 subject to limitations in one point of view, which in another 

 have no existence. As a function of ;:; it leaves x perfectly arbi- 

 trary, but as a function of x, and especially a periodic function, 

 it confines our attention to the ordinal character of z. The same 

 applies, perhaps, with still greater force to n, which in strictness 

 can be no other than a term, either affirmative or negative, in the 

 natural series of integers, agreeably to Mr. Herschell's definition 



(Examples, vol. ii. sect. 11). If rational fractions, as -, have 



appeared admissible, it was solely in virtue of the ordinal 



n 



character of the numerator. In fact, in representing vl/'" x = x 

 to be a periodic formula, we make two statements, viz. ^x =: ^x 



and x^r = x. Consequently ^J. ^ a; = %~^T = %""'' x. And 

 this is, in every respect, the prei'erable mode of operating, as 



Mr. Herapath will find, if he applies it to the case 4/ ^ x, when 

 a 



4.* X = X, which he has solved. (An.nah, Nov. 1824.) 



By reducing an irrational index to' converging fractions, we 



may render the function susceptible of indefinite approach toward 



the periodical state; e. g. /" x =.s x, is nearly solved by 

 f = %% when x^ X = x; more nearly b^vy = %^ when x' a; = j:; 

 still more nearly by^ = x^'' when %'' a: = x; and so on. But 

 such functions cannot with accuracy be termed periodical. 



An imaginary index destroys the co nditions of circulation, 

 being incapable of an ordinal character. 



Some caution would even appear to here tjaisite in interpolating 

 even at regular intervals, or the conditions of the problem may be 



completely altered ; as, for example, in ta king^f"" a; in Equation 

 (26) where a given effect is to recur only at- defijiite intervals. 



W. G. IIOKNEK. 



Article II. 

 On Mr. Horner*s Solution of ^r x = x. I 5y J. Herapath, Esq. 

 (To the Editors of the Annals of j Philosophy.) 



GENTLEMEN, i Uranford, March 1, 1826. 



You wiil, I have no doubt, allow me t ,0 correct a misrepre- 

 sentation in your present Number of a part ( )f my writings printed 

 in the Annuls for Nov. 1824. Mr. Horner, in your Number for 



