235 



variable will disappear from the result, except the effect of 

 those which are extremely nearly equal to the variable origi- 

 nally proposed. 



Poisson has made, with consummate skill, a great number 

 of applications of this method ; yet it appears to present, on 

 close consideration, some difficulties of the kind above alluded 

 to. In fact, the introduction of the system of factors, which 

 tend to vanish before the integration, as their indices increase, 

 but tend to unity, after the integration, for all finite values of 

 those indices, seems somewhat to change the nature of the 

 question, by the introduction of a foreign element. Nor is it 

 perhaps manifest that the original series, of which the sum is 

 indeterminate, may be replaced by the convergent series with 

 determined sum, which results from multiplying its terms by 

 the powers of a factor infinitely little less than unity ; while 

 it is held that to multiply by the powers of a factor infinitely 

 little greater than unity would give an useless or even false 

 result. Besides there is something unsatisfactory in employ- 

 ing an apparently arbitrary contrivance for annulling the 

 effect of those terms of the proposed series which are situated 

 at a great distance from the origin, but which do not them- 

 selves originally tend to vanish as they become more distant 

 therefrom. Nor is this difficulty entirely 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 differential 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 demonstra- 

 tion 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 



