Sir William Rowan Hamilton on Fluctuating Functions. 279 



M„ = (rrfa)^P„, (b") 



becomes infinitely often equal to a given constant, a, for values of a which extend 

 from negative to positive infinity, and are such that the interval between any one 

 and the next following is never greater than a given finite constant, b. With 

 these suppositions respecting the otherwise arbitrary function p„, the theorems 

 (z) and (z') may be expressed as follows : 



and 



b "" 



fx = -=f~' \ do.\ d^ P^(a_x)/a ; (or > a, < 6) (b) 



■u being determined by the equation 



CO 1^ 



^=\ da\d^V,^. (c") 



Now, by making 



p„ = cos a, 



(a supposition which satisfies all the conditions above assumed), we find, as 

 before, 



and the theorem (b) reduces itself to the less general formula (a), so that it 

 includes the theorem of Fourier. 



[10.] If we suppose that x coincides with one of the limits, a or h, instead 

 of being included between them, we find easily, by the foregoing analysis, 



/„ = ^^-'f*</afc//3p,,_„/„; (d") 



f,-^-'ida\d^v,,^_,,f^; ■ (e") 



in which 



^^ = '^da ^ rf/3 P,„ ; (f") 



•a 



