Let 

 then 



Sir William Rowan Hamilton on Fluctuating Functions. 281 



(2n-l-l)(a— ») 



f -1^ ^'™ C*^ So day, 



" So dav^ 



\ dav,— Q„,„\ dav,; (o") 



*'(2B — l)a *'o 



l+Qaa + Q».2 + ... + Q.» = ^-^ ^^' (P") 



So^« Pa 



and the formula (n") becomes 



/. = ^-' P„ (^* rfa/„ + 2(„)1 5* rfa Q_ .,„/„) ; (c) 



in which development, the terms corresponding to large values of n are small. 

 For example, when p,. = cos a, then 



w = TT, Po = 1, Q„,„ = 2 cos Ina, 



and the theorem (c) reduces itself to the following known result : 



/, = ^-' (J* flfa/„ + 2 2,„r.£ ^« COS (2«a - 2w^)/„) ; (q") 



in which it is supposed that x ^ a, x < b, and that h — o !J> x, in order that 

 a — X may be comprised between the limits ± tt, for the whole extent of the 

 integration ; and the function y^ is supposed to remain finite within the same 

 extent, and to vary gradually in value, at least for values of the variable a which 

 are extremely near to x. The result (q") may also be thus written : 



/. = -n-' 2(„;_:C ^«cos {2na - 2nx)f^ ; (r") 



'J a 



and if we write 



it becomes 



0v = ^ 2cn,- : J d^ COS (n(8 - ny) 0^ (s") 



the interval between the limits of integration relatively to /3 being now not 



VOL. XIX. 2 o 



