270 Sir William Rowan Hamilton on Fluctuating Functions. 



i 



"M + l 



6„ being a factor which cannot exceed the limits ±1. Hence, if we change 

 successively n io n-\-\,n-\-2, ..n-\-ni ~\, and add together all the results, 

 we obtain this other rigorous expression, for the integral of the product n^„ f<j 

 extended from a =z j3~' a„ to a = |3~* a„^m : 



\_ (^aN^„F„=0^-'(a„^,„-«,)c8; (y) 



'^ n 



in which 8 is the greatest of the m quantities 6„, 8„^j, ..., or is equal to one of 

 those quantities, and is not exceeded by any other ; and 6 cannot exceed ±: 1 . 

 By taking j3 sufficiently large, and suitably choosing the indices n and n-\-m, 

 we may make the limits of integration in the formula (y) approach as nearly as 

 we please to any given finite values, a and b ; while, in the second member of 

 that formula, the factor ^~' (a„ + „ — «„) will tend to become the finite quantity 

 h — a, and 6c cannot exceed the finite limits ±c ; but the remaining factor 8 

 will tend indefinitely to 0, as j8 increases without limit, because it is the difference 

 between two values of the function f,., corresponding to two values of the varia- 

 ble a of which the difference diminishes indefinitely. Passing then to the limit 

 ^ zr GO, we have, with the same rigour as before : 



■J) 

 da N,„ F„ = ; (z) 





which is the theorem that was announced at the end of the preceding article. 

 And although it has been here supposed that the function f„ receives no sudden 

 change of value, between the limits of integration ; yet we see that if this func- 

 tion receive any finite number of such sudden changes between those limits, but 

 vary gradually in value between any two such changes, the foregoing demonstra- 

 tion may be applied to each interval of gradual variation of value separately ; 

 and the theorem (z) will still hold good. 



[5.] This theorem (z) may be thus written : 



^ 



lim r* , ^ , ,\ 



= 00 3/«N^«F« = 0; (a) 



