308 Sir William Rowan Hamilton on Fluctuating Functions. 



developed according to cosines of the odd multiples of — , by means of the for- 



mula (y'"), which here becomes, by changing I to^, andy to p, 



Px = 2(„j, A,„_, cos ^ -^ , (a''"') 



in which 



4r'| (2w — l)7ra /uy/z/N 







the function n^: at the same time admitting a development according to sines of 

 the same odd multiples, namely, 



and the constant ts being equal to the following series, 



Thus, In the case of the last article, where jp = 2, and p„ = 1 from a =: to 

 a = 1, we have 



^"-'"tt 2«-1 ' ^^ > 



Px = -(^cos — — 3 'cos— -4-5 'cos— ...j; (f^"^) 



y. = -, (^sin Y - ^ ' sm — + 5 ^ sm -^ - ...j ; (g"-"') 



^ = -(1-^-3-^+5-^ — 7-'+"-); (h''''0 



so that, from the comparison of (w^^^) and (h^^^^), the following relation results : 







But most of the suppositions made in former articles may be satisfied, without 

 assuming for the function p the periodical form assigned by the conditions (t^^). 



