236 



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 function, such as the 

 sine or cosine of an infinite multiple of an arc, changes sign 

 infinitely 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 

 afterwards integrated between any finite limits ; the integral 

 of the product will be zero, on account of the mutual destruc- 

 tion 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 vari- 

 able 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 published 

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

 theorem called by his name ; since it exhibits the actual pro- 

 cess, one might almost say the interior 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 construc- 

 tions, 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 Poisson 

 has given on the subject of these transformations. For ex- 

 ample, although Fourier, in the proof just referred to, of the 



