284 Sir William Rowan Hamilton on Tluctuating Functions. 



becomes 



/, = TT-' C da\d^ COS (/3a — /3a;)/„ (a'") 



which is a more usual form than (a) for the theorem of Fourier. In general, 

 the deduction in the present article, of the theorem (d) from (c), may be regarded 

 as a verification of the analysis employed in this paper, because (d) may also be 

 obtained from (b), by making the limits of integration infinite ; but the demon- 

 stration of the theorem (b) Itself, in former articles, was perhaps more completely 

 satisfactory, besides that it involved fewer suppositions ; and it seems proper to 

 regard the formula (d) as only a limiting form of (b). 



[13.] This formula (d) may also be considered as a limit in another way, by 

 introducing, under the sign of integration relatively to /3, a factor f^^ such that 



F„=l, F^=0, (b'") 



in which k is supposed positive but small, and the limit taken with respect to It, 

 as follows : 



/- = A; = ' '^~' \ ^" (^ ^^ P^f— ) ^*^)/"- (^) 



It is permitted to suppose that the function f decreases continually and gradually, 

 at a finite and decreasing rate, from 1 to 0, while the variable on which it 

 depends increases from to oo ; the first differential coefficient f' being thus 

 constantly finite and negative, but constantly tending to 0, while the variable is 

 positive and tends to cc. Then, by the suppositions already made respecting the 

 function p, if a — or and k be each different from 0, we shall have 



\ c?^P^(a_x)F*^ = Ft^N^(„_^, (a — or) ' 

 — k{a—x) 'V flf/3N^(„_,)F'i^; 



(C'") 



and therefore, because f^ = 0, while n is always finite, the integral relative to j8 

 in the formula (e) may be thus expressed : 



m 



\ «?^P^(„_x)Fi^ = (a — ar)-'i|ri_,(„_^„ (d'") 



the function ^ being assigned by the equation 



