Sir William Rowan Hamilton on Fluctuating Functions. 301 



a zz. V 



and p„ will be different from 0, because the real roots of the equation (a''0 have 

 been supposed unequal. Conceive also that the integral 



QO 



tends to some finite and determined limit, which may perhaps be different for 

 different roots v, and therefore may be thus denoted, 



as j3 tends to oo , after the given law referred to at the end of the last article. 

 Then, by writing 



and supposing j3 to be very large, we easily see, by reasoning as in former articles, 

 that the part of the integral 



which corresponds to values of a — .r in the neighbourhood of the root v, is very 

 nearly expressed by 



and that this expression is accurate at the limit. Instead of the equation (s), we 

 have therefore now this other equation : 



2. W, PT'/x + v = V . \ da S„_:r,;s/„ ; (t) 



the sum in the first member being extended to all those roots v of the equation 

 (a^^), which satisfy the conditions 



x-\-v>a,<b. (k^O 



If one of the roots v should happen to satisfy the condition 



x-\-v = a, {V) 



the corresponding term in the first member of (t) would be, by the same princi- 

 ples, 



