286 Sir William Rowan Hamilton on Fluctuating Functions. 



is a cosine, and f a negative neperian exponential, so that 



p„ = cos a, F„ = e~% 

 then, making \ = A;"' (a — x), we have 



C rfjS e-'^ cos (|3a — /ar) = (a - t)-' -^^ ; 







» - 



V-x = Wy e"^ sin X7 = — — - ; 

 and 



It is nearly thus that Poisson has, in some of his writings, demonstrated the theo- 

 rem of Fourier, after putting it under a form which differs only slightly from the 

 following : 



lim f* (* 



/. = 7r-^^^^^^rfaJ^rf^e-*^COs(|3a-j3ir)/; (1'") 



namely, by substituting for the integral relative to /3 its value 



k 

 1^ -\- {a — xf ' 



and then observing that, if k be very small, this value is itself very small, unless 

 a be extremely near to x, so that f^ may be changed tof^ ; while, making 

 a=z x-\- k\, and integrating relatively to \ between limits indefinitely great, the 

 factor by which this function y^, is multiplied in the second member of (1'"), is 

 found to reduce itself to unity. 



[14.] Again, the function f„ retaining the same properties as in the last 

 article for positive values of a, and being further supposed to satisfy the condition 



F_. = F„, (m'") 



while k is still supposed to be positive and small, the formula (d) may be pre- 

 sented in this other way, as the limit of the result of two integrations, of which 

 the first is to be effected with respect to the variable a : 



