SiE William Rowan Hamilton on Fluctuating Functions. 265 



[1.] The theorem, discovered by Fourier, that between any finite limits, 

 a and b, of any real variable x, any arbitrary but finite and determinate function 

 of that variable, of which the value varies gradually, may be represented thus, 



1 (** C® 

 fx zz -\ da\ d^cos (/3a — Px)/a, (a) 



with many other analogous theorems, is included in the following form : 



/x = \ da\ dp(f)(x,a,^)fa; (b) 



the function being, in each case, suitably chosen. We propose to consider 

 some of the conditions under which a transformation of the kind (b) is valid. 

 [2.] If we make, for abridgment, 



^|r{x,a,p) = \ c?p0(ar,a,/3), (o) 



the equation (b) may be thus written : 



Jx =:\ dayjf (x, a, <x)fa. (d) 



This equation, if true, will hold good, after the change of/a, in the second 

 member, to/a -\- va ; provided that, for the particular value a = a?, the additional 

 function Fa vanishes ; being also, for other values of a, between the limits a and 

 h, determined and finite, and gradually varying in value. Let then this func- 

 tion F vanish, from a = a to a = \, and from a=:/xto a^6; \ and jjl being 

 included, either between a and x, or between x and h ; so that x is not included 

 between \ and fi, though it is included between a and b. We shall have, under 

 these conditions, 



0=\ (/a 1^ (x, a, go) Fa; (e) 



the function f, and the limits \ and fi, being arbitrary, except so far as has 

 been above defined. Consequently, unless the function of a, denoted here by 

 ■^ (or, a, 00 ), be itself = 0, it must change sign at least once between the limits 

 azz\ a=: n, however close those limits may be ; and therefore must change 

 sign indefinitely often, between the limits a and x, or x and b. A function 



VOL, XIX. 2 m 



