the Surn^ of Neutral arid Periodic Series. ill 



of these, in extreme cases, we disconnect those cases from the 

 others, remove from them a condition which they must other- 

 wise have obeyed, and regard them as isolated and indepen- 

 dent. For example, the isolated case of the general series dis- 

 cussed above, which arises from putting A= 1, is 



h cos 5 + cos 2 5 + cos 3 5 + ... cos n 9, 



of which tlie correct sum is the expression [2.] above, for[l.] 

 vanishes^ that is, the sum is 



cos (» + 1)9 — cos n d ^ 

 2'( f— cos d) ' 



and it is the error committed in confounding this isolated and 

 independent case with that which is really the limit of the ge- 

 neral series, and therefore under the control of the law of con- 

 tinuity : — it is this error (and errors such as this) that has led 

 to the additional error of supposing sin oo = and cos co = 0, 

 since it was found that the sums of the two series, supposed to 

 be identical, would really differ in the case of « = co , unless 

 the sine and cosine of an infinite arc v/ere made equal to zero. 

 In some instances cos oo is assumed to be 1 instead of 0, 

 an assumption which, like that just noticed, seems to be ne- 

 cessary in order to prevent contradiction in the results of cer- 

 tain definite integrations ; which integrations however will be 

 found to involve the same error as that noticed above in re- 

 ference to series ; the error, namely, of bringing an isolated 

 expression under the dominion of an arbitrary law, and then 

 reasoning upon it in reference to that law. Although, as stated 

 above, it is not my present intention to consider at any length 

 the influence of this error upon the existing theory of definite 

 integrals, yet it may be proper to adduce an example of it. 

 It is easily proved that 



/•* a /*°° 1 



/ e-''''cosxd.r = 5, / e~"'^sinjrr7.r =-— — 5, 



from which it certainly follows, though the inference is denied 

 by Mr. Moon, that in the limit, when a = 0, the true values 

 of these integrals are and 1. Yet it is not true, as Poisson 

 and his followers affirm, that, in virtue of this, .fia^i 



/ cos X dx ss 0, / smx dx =^ \i 



inasmuch as these and the limits of the foregoing general 

 forms are very different things. I ventured a conjecture above, 

 that Poisson countenances this important error in his memoirs 

 on series and definite integrals in the Journal of the Polytech- 

 Phil. Mag. S. 3. Vol. 27. No. 182. Dec. 184.5. 2 G 



