IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 435 



ever — suspected by Mr. De Morgan. But it is proper to state that Poisson nowhere countenances 

 this notion ; nor is it implied in his principles, though it has been thought to follow from them. 

 It is true that Poisson makes 



/ dx sin .V = 1, and / d,v cos x = 0. 



It is also true that these integrals are respectively I — cos cc , and sin e© : but it does not follow 

 that we have any right to equate Poisson's results with these. Poisson virtually selects a particular 

 value out of the innumerable values of cos ce ; and a particular value out of the innumerable values 

 ■ of sin CO ; these selected values are each zero. He does not deny the existence of the other values, 

 nor say that sin zc and cos eo are zero only, as others have said : he expressly declares that he 

 takes that particular value of cos co which unites in continuity with the values of 



/ d,v e "' sin x ; 



and that particular value of sin w which unites in continuity with the values of 



d,r e~ "' cos a.; 



J 



it being understood that a varies from some superior value down to zero ; and his doctrine is that, 

 by taking the extreme limit thus reached, he gets, in each case, " une valeur unique qu'on pent 

 employer dans I'analyse." The fault of Poisson consists solely in his bringing indeterminate 

 expressions under the control of arbitrary conditions, in virtue of which that indeterminatcness 

 is destroyed, and unique values deduced ; and in consequence of which these unique values — as in 

 the instance of the series 1 — 1 + 1 — 1 + &c. — are frequently not even among the indeterminate 

 .set: but this great man must not he charged with the palpable error of making the sine and cosine 

 of an infinite arc zero*. It should also, in justice to the same illustrious analyst, be observed 

 further, that some English authors, under the impression that they have been carrying out Poisson's 

 views, have also, on other points, employed reasonings, and arrived at conclusions, which those views 

 do not justify. The results which Poisson assigns to the integrations noticed in this paper are all 

 frtie as far as (hey go. He chooses one out of an infinite variety of equally admissible values, and 

 disregards all the others : — a fault which appears to me to be analogous to that which would 

 he committed by arbitrarily selecting one of the n roots of an equation of the w"' degree, to he 

 employed in physical applications, and rejecting all the others But, from a pretty careful exami- 

 nation of Poisson's different Memoires on Series and Definite Integrals, I can find no foundation for 



the statement recently made, that "Poisson would admit 1° — 2" + 3'- - 4' + = O." He rejects 



diverging series : and in applying his principles to cases where divergency might be suspected, he 

 takes care, in order to justify his mode of proceeding, to remove the suspicion, by showing that the 

 series must be convergent. (See Thiorie de In Chxileur, p. 188.) 



Resuming now the consideration of the definite integrals, I have to remark, that among those 

 that have been rejected are 



Jf" sin ffl.p fco&ax 

 dm and / d.r ; 

 .1' -^ 1 + x' 



the grounds of this rejection being that these integrals have not the values hitherto assigned 



♦ " I,e« ninui et ciiKinus il'uti arc inlini nonl ividemment <lf» p(!rio(liquc, que s'cicndeiu ii I'inlini : ce» iiiti'KraU n'onl uusm iU» 



<)uanlil^« inilelcrmincej." I'oi««on ; Journal de V Ecole I'olytech. valeurn dt'termini-ei., que ([uand on Ic» rcgarde commc Ics liuiiie« 



C«h. XIX. p. 407. d'autres inte^'ralcH, dont lc» c'li<men» convergent vern lero, et «oni 



"La nianiere dont nous avons cont»iddr(* le« Iii5rie8 p(5riodique , nul» a rinlini." /6i(/. , p. 41*1. 

 infiniei, f'appliquc i^galcnicnt aux \nl(gta,\t di<tinies de quantities 



