Sir "William Rowan Hamilton on Fluctuating Functions. 285 



For any given value of A, the value of this function ^ is finite and determinate, 

 by the principles of the sixth article ; and as \ tends to oo, the function i^ tends 

 to 0, on account of the fluctuation of n, and because f' tends to 0, while 7 tends 

 to GO ; the integral (d'") therefore tends to vanish with k, if a be different from 

 X ; so that 



lim 



k 



™0-J (//3p„„_,f., = 0, ifa>ar. (f") 



On the other hand, if a = or, that integral tends to become infinite, because we 

 have, by (b'"), 



Thus, while the formula (d'") shows that the integral relative to /3 in (e) is a 

 homogeneous function of a — x and k, of which the dimension is negative unity, 

 we see also, by (f") and (g"')> that this function is such as to vanish or become 

 infinite at the limit A; = 0, according as a — :r is different from or equal to zero. 

 When the difference between a and x, whether positive or negative, is very small 

 and of the same order as k, the value of the last mentioned integral (relative to 

 /3) varies very rapidly with a ; and in this way of considering the subject, the 

 proof of the formula (e) is made to depend on the verification of the equation 



00 



z^-'C dX^^\-'=\. (h'") 



But this last verification is easily effected ; for when we substitute the expression 

 (e'") for ^„ ai^d integrate first relatively to X, we find, by (r'), 



oo 



C rf\N,,\-' = ^; (i'") 



it remains then to show that 



- f rf7 f; = 1 ; (k"') 



and this follows immediately from the conditions (b'"). For example, when p 



