294 Sir William Rowan Hamilton on Fluctuating Functions. 



tlon of a, and to have its mean value equal to zero, as p tends to become infinite, 

 for all values of « and a; which are different from each other, and are both com- 

 prised between the limits of the integration relative to a ; in such a manner as to 

 satisfy the equation 



J^^«^.„.»/„ = 0, (k) 



which is of the form (e), referred to in the second article ; provided that the 

 arbitrary functionyis finite, and that the quantities \, /i, x, a, are all comprised 

 between the limits a and b, which enter into the formula (h) ; while « is, but x 

 is not, comprised also between the new limits A and jjl. But when a.-=^ x, the 

 sum (i) tends to become infinite with p, so that we have 



■fx,,.« = co, (l) 



and 



\ d<^i;.a.^fa=A., (m) 



e being here a quantity indefinitely small. For example, in the particular ques- 

 tion which conducts to the development (y'"), we have 



2 



0;,,^p = J- cos px cos pa, (\"') 



and 



(2ra — l)7r 

 P = 2? ' ("^^ 



therefore, summing relatively to p, or to n, from w = 1 to any given positive 

 value of the integer number n, we have, by (i), 



. mr (a — x) . mr(a4-x) 



sm ^ sm — ^-j-^ — - 



and it is evident that this sum tends to become a fluctuating function of a, and to 

 satisfy the equation (k), as p, or n, tends to become infinite, while a, and x are 

 different from each other, and are both comprised between the limits and l. 

 On the other hand, when a becomes equal to x, the first part of the expression 



