1824.] Mr. Herapath on the Solution of^ n x = x. 421 



frequently preferable to either of the preceding formula; in 

 developing algebraic values of vj/ v x. 



These very simple expressions at once give several of the 

 properties of the functional root. Thus from the manner in 

 which k is introduced, it may be any integral number 0, 1,2, 3, 

 to infinity ; and therefore, generally, ^ x will have k functional 



roots, k being the least integer that will make - an integer. If, 



therefore, n be an integer, the number of roots will be n ; if a 

 fraction, they will be equal to the numerator of the fraction ; 

 and if an irrational number they will be infinite. And since A; 

 may be = o, it follows that whatever be the values of n and v, 

 one value of \J/ x is x. It should however, in the solution of 

 problems, be carefully considered, whether the conditions of the 

 problem will admit the property \J/" x = x when vj/ n x = x. A 

 very skilful mathematician in this branch of analysis, has fallen 

 into error in the solution of one of his problems from not attend- 

 ing to this. 



Again, because either expression (25) or (26) is symmetrical 

 with respect to v and k, if k remain constant, and 0, 1,2, .... 

 be substituted for v, the results will be the same as if v remained 

 constant, and 0, 1, 2, 3, .... be substituted for k. Consequently 

 whatever be the value of», the several values of \f/ x will be equal 



to \J/° x, \J/ x, \J/ 2 x, \[/ 3 x, This conclusion Mr. Babbage has 



arrived at by a very different process in his paper in the Philos. 

 Transac. for 1817, when n is an integer ; but such a limitation 

 is evidently unnecessary. 



Let us assume our expression (25) or (26) to be a function of 

 k, v, and n. Then 



supposing that in the former instance ^" x = x, and in the latter 

 4/,"' x = x. If therefore n — p n, 



♦"-/(£)-/(£)-*- 



and V x = ^, 2 x, tf* x = i> 3 x, ^' ,p x = $* x, 



But if/) be an integer, ^ p x, V x, .... are, by what we have 

 just shown, functional roots of 4> H x = x. Therefore and because 

 ij,, x, \J>,- x, .... are the functional roots of i^ c "« x = x, it fol- 

 lows, that if the order of a periodic function be an inteo-ral 

 multiple of the order of another periodic, the roots of the latter 

 are roots of the former, whatever be the values of the orders of 

 periodicity. Mr. Babbage has drawn, in the above-mentioned 

 paper, the same inference for integral orders. 



If in vj/" x = x we substitute -^~" for .r we have 



ij,-" x= x; 



