j'rfa5"#P^/„ = .2.-^, (X-) 



Sir William Rowan Hamilton on Fliictuating Functions. 315 



7« J_„ «— a;' ^ ' 



in which /3 is indefinitely large and positive, and the differential coefficient 7'^ of 

 the function 7 is supposed to be finite, and different from 0. A little considera- 

 tion shows that the integral in this last expression is = it w, -cr being the same 

 constant as in former articles, and the upper or lower sign being taken according 

 as 7'x is positive or negative. Denoting then by 1/7' x^ the positive quantity, 

 which is = + 7'a; or = — 7'^, according as 7'^ is > or < 0, the part (v^^) 

 of the integral (t^^) is 



-5^5 (w") 



and we have the expression 



^ J^ 



the sum being extended to all those roots x of the equation (u^^) which are > a 

 but < b. If any root of that equation should coincide with either of these 

 limits a or h, the value of a in" its neighbourhood would introduce, into the se- 

 cond member of the expression (x^^), one or other of the terms 



7a 7a 7» 7» 



the first to be taken when 7^ = 0, 7'a > ; the second when y^ = 0, y'a < ; 

 the third when 7^ =0, 7'^ > ; and the fourth when 74 = 0, 7'j < 0. If, 

 then, we suppose for simplicity, that neither 7„ nor 74 vanishes, the expression 

 (x^'^) conducts to the theorem 



2./x = ^-' ( rfa C dp P,y„ /7J ; (X) 



•^a »^o 



and the sign of summation may be omitted, if the equation 7* = have only one 

 real root between the limits a and b. For example, that one root itself may then 

 be expressed as follows : 



X=zr-'^ da^ dp P^ a VT?. (z«) 



The theorem (x) includes some analogous results which have been obtained by 

 Cauchy, for the case when p is a cosine. 



2 s 2 



