278 Sir William Rowan Hamilton on Fltcctuafing Functions. 



and therefore, for all such values of ^, 



B e^ = const. 



The constant in this last result is easily proved to be equal to the quantity a, 



by either of the two expressions already established for that quantity ; we have 



therefore 



B =: a e~^, 



however little the value of /3 may exceed ; and because b tends to the limit - 

 as ^ tends to 0, we find finally, for all values of /3 greater than 0, 



These values, and the result 



\ 



J sm a 



da ^: -n. 



to which they immediately conduct, have long been known ; and the first relation, 

 above mentioned, between the integrals a and b, has been employed byLEGENDRE 

 to deduce the former integral from the latter ; but it seemed worth while to 

 indicate a process by which that relation may be made to conduct to the values 

 of both those integrals, without the necessity of expressly considering the second 

 differential coefficient of b relative to /3, which coefficient presents itself at first 

 under an indeterminate form. 



[9.] The connexion of the formula (z') with Fourier's theorem (a), will be 

 more distinctly seen, if we introduce a new function p„ defined by the condition 



N„ = J"rfaP„, (a") 



which is consistent with the suppositions already made respecting the function n„. 

 According to those suppositions the new function p„ is not necessarily continuous, 

 nor even always finite, since its integral n„ may be discontinuous ; but p„ is sup- 

 posed to be finite for small values of a, in order that n„ may vary gradually for 

 such values, and may bear a finite ratio to a. The value of the first integral of 

 p. is supposed to be always comprised between given finite limits, so as never to 

 be numerically greater than ± c ; and the second integral. 



