426 Mr. llerapatli on the Solution of ty 1 x = x. [Dec. 



c being a constant. By changing x into -, it appears when 



n = — 1, that c can only be + 1 or — 1. If, however, Lap- 

 lace's method of differences be followed, we have for the solution 



of (37) 



log 2 A„ — log* x 



^ x = A, c k^ — " (38) 



A , A, being the constants of integration. If now we change x 

 into -, the solution becomes, putting p for the former exponent, 



4,- = A, c v ~* 



X 



Multiplying this by c, we have 



c^/i = A, c" = $x, 



X 



which, therefore, satisfies the conditions of equality of the ques- 

 tion without any apparent limitation to c, that is, without the 

 necessary limitation the question requires. A part only, there- 

 fore, of the conditions of the question are satisfied ; and this 

 arises from the periodicity of the exponent being destroyed by 

 the development of p. If we consider that by changing x into 



- twice successively, p returns into itself, and suppose p by one 

 such change to become q, we may easily find that 



c?+ a = c"or c = +1. 

 And in the same way if n = \/ 1, we should have c = y J, 

 provided the value of c be sought by the non-development of the 

 function. 



These illustrations will, I hope, obviate the difficulty noticed 

 by Mr. Babbage in Phil. Trans. 1817, without having recourse 

 to the ingenious but, I presume, controvertible idea of the func- 

 'tion -\> x, having simultaneously different values in different parts 

 of the same equation. The same may be said of Mr. Herschel's 

 views, p. 120, vol. ii. of the Examples. 



As this is a subject of considerable importance in the 

 theory of functions, J shall here briefly notice another more 

 general instance of my observation ; namely, that the same 

 expression may be periodic or non-periodic according to its 

 form. In our solution (25), k and <f> are perfectly arbitrary. 

 Take then $ * = sin -1 .r, and we have 



i „ •_!• (2 k v K , •_, • ) 2 k v K , /nrx. 



vj/ x = sin sin i + sin sina;( = t-x . . . . (oy) 



the former value of which is periodic for every value of n, pro- 

 vided sin and si?i~ l act separately and distinctly, but the latter 

 is not. I am aware it will be contended that the arithmetical 

 values of 4'" x in (39) when differently developed, are not neces- 

 sarily the same. This may be true in the present case, but it 





