Sir William Rowan Hamilton on Fluctuating Functions. 295 



n 



(x^'') becomes = j, and therefore tends to become infinite with n, so that the 



equation (l) is true. And the equation (m) is verified by observing, that if 

 or > 0, < /, we may omit the second part of the sum (x^^), as disappearing in 

 the integral througli fluctuation, while the first part gives, at the limit, 



mr (a — s) 



sm- 



2/sm-4^ 



If X be equal to 0, the integral is to be taken only from to e, and the result is 

 only half as great, namely, 



. mra. 

 sin— J- 



but, in this case, the other part of the sum (x^^) contributes an equal term, and 

 the whole result is^g. If x =.1, the integral is to be taken from / — e to /, and 

 the two parts of the expression (x^'') contribute the two terms ^y^ and — ^y), 

 which neutralize each other. We may therefore in this way prove, d posteriori, 

 by the consideration of fluctuating functions, the truth of the development (y'") 

 for any arbitrary but finite function y^j and for all values of the real variable x 

 from X ^0 to s=: I, the function being supposed to vanish at the latter limit ; 

 observing only that if this function/*^ undergo any sudden change of value, for 

 any value x'^ of the variable between the limits and /, and if x be made equal 

 to ar" in the development (y'")> the process shows that this development then 

 represents the semisum of the two values which the function y receives, imme- 

 diately before and after it undergoes this sudden change. 



[18.] The same mode of a posteriori proof, through the consideration of fluc- 

 tuating functions, may be applied to a great variety of other analogous develop- 

 ments, as has indeed been indicated by Fourier, in a passage of his Theory of 

 Heat. The spirit of Poisson's method, when applied to the establishment, a 

 posteriori, of developments of the form (h), would lead us to multiply, before the 

 summation, each coefficient 0^„_p by a factor Fk,^ which tends to unity as k tends 



