Sir William Rowan Hamilton on Fluctuating Functions. 313 



If" 



7^ = -\ da cos (2/3 sin a), (g") 



when /3 Is a large positive quantity, we need only attend to values of a which 



differ little from - ; making then 



sin a =: 1 — J/*, da 



__^dy 



(h«) 



v/2— y' 



and neglecting y^ in the denominator of this last expression, the integral (g^^) 

 becomes 



y^ = A^cos2^+B^sin2^, (i^*) 



in which, nearly, 



*^ = — i /^cos(2^^/^) = -_=; 



v/2 7r/3 

 B, = ^L.^sin(2^y) = ^; 



^ 



(k") 



so that the large values of ^ which make the function (g") vanish are nearly of 

 the form - 



n-n TT 

 2"~8' 



(1-) 



n being an integer number ; and such is therefore the approximate form of the 

 large roots a„ of the equation (p'^^^O • results which agree with the relations 

 (q''^^^), and to which Poisson has been conducted, in connexion with another sub- 

 ject, and by an entirely different analysis. 



The theory of fluctuating functions may also be employed to obtain a more 

 close approximation ; for instance, it may be shown, by reasonings of the kind • 

 lately employed, that the definite integral (g^^) admits of being expressed (more 

 accurately as j8 is greater) by the following semiconvergent series, of which the 

 first terms have been assigned by Poisson : 



/,= ;^2,,UO]-n[-^]0W)-^cos(2i3_^-j); (m-) 



and in which, according to a known notation of factorials. 



vol. XIX. 



2s 



