296 Sir William Rowan Hamilton on Fluctuating Functions. 



to 0, but tends to vanish as p tends to co ; and then instead of a generally/ fluc- 

 tuating sum (i), there results a generally evanescent sum (k being evanescent), 

 namely, 



2pFA,,0^_„_, = Xx,a,*:,p» (n) 



which conducts to equations analogous to (k) (l) (m), namely, 



;-^™o5'rf-Xx.a...-/» = 0; (o) 



^!!"nX..x,...=ao; (p) 



k = 



lim -'+• 



k 



%^^_dax.,.,.,.f.=f.. (q) 



It would be interesting to inquire what form the generally evanescent function 

 X would take immediately before its vanishing, when 



F*.. = «'*" 



and 



2p sm px sin pa 



''' pi — sin pi cos pV 



p being a root of the equation 



pi cotan pi = const. , 



and the constant in the second member being supposed not greater than unity. 



[19.] The development (c), which, like (h), expresses an arbitrary function, 

 at least between given limits, by a combination of summation and integration, was 

 deduced from the expression (m") of the eleventh article, which conducts also to 

 many other analogous developments, according to the various ways in which the 

 factor with the infinite index, n«(„_x)) May be replaced by an infinite sum, or 

 other equivalent form. Thus, if, instead of (0"), we establish the following equa- 

 tion, 



\ rfap„=: R„„V rfap„, (a'') 



♦^(2n_2)o •^0 



we shall have, instead of (c), the development : 



