Sir William Rowan Hamilton on Fluctuating Functions. 267 



than 0, the number of changes of sign of the function n^„, within this range of 

 the variable a, will increase indefinitely with /3. Passing then to the extreme or 

 limiting supposition, /3 = oo , we may say that the function n„„ changes sign 

 infinitely ofien within a finite range of the variable a on which it depends ; and 

 consequently that it is, in the sense of the last article, a fluctuating function. 

 We shall next consider the integral of the product formed by multiplying toge- 

 ther two functions of a, of which one is N^„, and the other is arbitrary, but finite, 

 and shall see that this integral vanishes. 



[4.] It has been seen that the function n„ changes sign at least once between 

 the limits a:=an, a=:anj^y Let it then change sign k times between those limits, 

 and let the k corresponding values of a be denoted by a„ ,, a„ j, ... o^, 4. Since 

 the function n,. may be discontinuous in value, it will not necessarily vanish for 

 these k values of a ; but at least it will have one constant sign, being throughout 

 not < 0, or else throughout not > 0, in the interval from a = a„ to a = a„ , ; it 

 will be, on the contrary, throughout not > 0, or throughout not < 0, from a„^ 

 to a„,2 ; again, not < 0, or not > 0, from a„ ^^ to a„ 3 ; and so on. Let then n„ 

 be never < throughout the whole of the interval from a„ ; to a„i^, ; and let 

 it be > for at least some finite part of that interval ; i being some integer 

 number between the limits and k, or even one of those limits themselves, pro- 

 vided that the symbols a„o, a„i^jare understood to denote the same quantities 

 as a„, Onj^y Let F„ be a finite function of a, which receives no sudden change of 

 value, at least for that extent of the variable a, for which this function is to be 

 employed ; and let us consider the integral 





c?a N„F„. (1) 



Let f' be the algebraically least, and f^^ the algebraically greatest value of the 

 function f„, between the limits of integration ; so that, for every value of a 

 between these limits, we shall have 



F„ — f' <: 0, f'' — F„ < ; 



these values f^ and f^', of the function f„, corresponding to some values d„i and 

 a\i of the variable a, which are not outside the limits a^i and 0^,1 + 1- Then, 

 since, between these latter limits, we have also 



2m2 



