274 Sir William Rowan Hamilton on Fluctuating Functions. 



increases indefinitely, because the part which corresponds to values of a between 

 any given value of the form o^^.^ and any other of the form a„4.„+p is included 

 between the limits ± ^ be f„ , which limits approach indefinitely to each other 

 and to 0, as m increases indefinitely. And in the integral (1'), if we suppose the 

 lower limit a to lie between a„_, and a„, while the upper limit, instead of being 

 infinite, is at first assumed to be a large but finite quantity b, lying between a„^„ 

 and a„_^™_^„ we shall only thereby add to the integral (o') two parts, an initial and 

 a final, of which the first is evidently finite and determinate, while the second is 

 easily proved to tend indefinitely to as m increases without limit. The integral 

 (1') is therefore itself finite and determined, under the conditions above supposed, 

 which are satisfied, for example, by the function f„ = ar\ if a be > 0. And 

 since the suppositions of the last article render also the integral 



\ rfaN^o"* 



determined and finite, if the value of a be such, we see that with these supposi- 

 tions we may write 



w = C C?aN„a~S (p') 



w being itself a finite and determined quantity. By reasonings almost the same 

 we are led to the analogous formula 



w-=C " day^a-'; (q') 



and finally to the result 



,^ = 70-^ + TU-" = C rfaN<.a-i; (r') 



in which w' and zs- are also finite and determined. The product (k') is there- 

 fore itself determinate and finite, and may be represented by zs/^. 



[7.] We are next to introduce, in (i'), the variable part of th^ function y^ 

 namely, 



which varies from/*a;_„ tofx+^i while y varies from — ^ to + 17, and in which 

 € may be any quantity > 0. And since it is clear, that under the conditions 



