Sir William Rowan Hamilton on Fluctuating Functions. 319 



removed, when Integration by parts is had recourse to, in order to show that the 

 effect of these distant terms is insensible in the ultimate result ; because it then 

 becomes necessary to differentiate the arbitrary function ; but to treat its diffe- 

 rential coefficient as always finite, is to diminish the generality of the inquiry. 



Many other processes and proofs are subject to similar or different difficulties; 

 but there is one method of demonstration employed by Fourier, in his separate 

 Treatise on Heat, which has, in the opinion of the present writer, received less 

 notice than it deserves, and of which it is proper here to speak. The principle 

 of the method here alluded to may be called the Principle of Fluctuation, and 

 is the same which was enunciated under that title in the remarks prefixed to this 

 paper. In virtue of this principle (which may thus be considered as having 

 been indicated by Fourier, although not expressly stated by him), if any func- 

 tion, such as the sine or cosine of an infinite multiple of an arc, changes sign in- 

 finitely often within a finite extent of the variable on which it depends, and has 

 for its mean value zero ; and if this, which may be called a fluctuating function, 

 be multiplied by any arbitrary but finite function of the same variable, and after- 

 wards Integrated between any finite limits ; the integral of the product will be 

 zero, on account of the mutual destruction or neutralization of all its elements. 



It follows immediately from this principle, that if the factor by which the 

 fluctuating function is multiplied, instead of remaining always finite, becomes 

 infinite between the limits of integration, for one or more particular values of the 

 variable on which it depends ; it is then only necessary to attend to values in the 

 immediate neighbourhood of these, in order to obtain the value of the integral. 

 And in this way Fourier has given what seems to be the most satisfactory pub- 

 lished proof, and (so to speak) the most natural explanation of the theorem called 

 by his name ; since it exhibits the actual process, one might almost say the in- 

 terior mechanism, which, in the expression assigned by him, destroys the effect of 

 all those values of the auxiliary variable which are not required for the result. 

 So clear, indeed, is this conception, that it admits of being easily translated into 

 geometrical constructions, which have accordingly been used by Fourier for that 

 purpose. 



There are, however, some remaining difficulties connected with this mode of 

 demonstration, which may perhaps account for the circumstance that it seems 

 never to be mentioned, nor alluded to, in any of the historical notices which 



