1824.] Mr. Herapath on the Solution of fi = x. 425 



consequences that can in any way flow from them, are naturally 

 comprehended in the arbitrary function of the direct solution. 



I cannot refrain here, while on the subject of periodic func- 

 tions, from mentioning two curious cases of periodic solutions 

 which are perfectly successful when the operations are merely 

 indicated but not performed, and yet fail when developed ; 

 though the developed and undeveloped values are the same. 

 The first I shall mention was noticed in elucidating some diffi- 

 culties in this calculus to my promising friend and pupil Mr. 

 Mervyn, Crawford, and the next occurred while considerino- the 

 source of what Mr. Babbage denominates " a very difficult 

 subject." It was mentioned to him in a letter dated March 22, 

 1824. 



Let \[/ x be a periodic of the second order whose solution is 

 evidently 



F (x, ^ x) = 0, 



where the form of F is to be determined. Substitute 4- x for x, 

 and it becomes 



F {+ x, V~ x] = F {* x, x} = o = F {.r, 4, x} 



F is, therefore, symmetrical with respect to x and 4- x\ Conse- 

 quently a solution is, 



•b x n + x n = 0, 



in which n is unlimited, or 



4, x = v - x " = x • y t» l (36) 



Now in all cases the first of these values V — x n answers the 



conditions of the question ; for 4^ x = V — (V — x n )" = 

 \/ x" = x. But in the second value x . V — 1, we have 



^x = x.(- If,- 



which if n be of the form, 2 p becomes %^ 2 a; = j: . !y — 1, an 

 expression that cannot, with any integral value at least to p, be 

 = x. It may be asked, in what this unexpected anomaly con- 

 sists ? The answer, I conceive, is obvious, if we seek it from the 

 nature of the functions in question. Periodic functions are 

 algebraic expressions whose property of periodicity depends not 

 on the value but the form of the expression. If, therefore, the 

 value be the same but the form be changed, the expression may 

 no longer be periodic. Thus it happens in the above instance, 

 the values of the two expressions are the same, but the forms 

 different — the negative sign in the one case being a mere sign oi 

 subtraction, and in the other a factor. 



This reasoning will appear still clearer in the following case. 



Suppose we have the equation 



$ x — c $ ,x % (37) 



