the Sums of Neutral and Periodic Series. :4|B9 



;,.y. This rule cannot, as usually supposed, apply lo series that 

 lire strictly neutral ; that is, not connected with more general 

 forms by the principle of continuity ; nor can it apply even 

 where such connexion exists, unless, in virtue of that con- 

 nexion, the terms tn+n ^n+25 &c. become zero. It cannot 

 apply therefore to the series 



cos w — cos 2 a; + cos Sx— cos 4*0; + &c. 

 when it passes into iiiri g'iiiijoima io 



1 _ 1 + 1 _ 1 + 1 _ &c., "" ■ '^ '-- ^ 



^iibr to 



'^'^'a + ^cos^ — ccos2^ + acosSx + 5cos4^ — ccos5;j7 + &c.y 

 in which c = a + 6, when it passes into 



a + b — c + a + b — c+ &c. ; 

 and this consideration alone is sufficient to show the fallacy 

 of asserting the general value of the first of these series to 



be 4. '*'"''' 



Let us now examine the series ''fiy* -H ? jIj .r. 



— + A cos d + A^ cos 2 fl + A^ cos 3 9 + ... A" cos n $, 



so intimately connected with Fourier's integral, and which 

 has already been the subject of consideration in Mr. Moon's 

 paper before adverted to. This series, as there shown, or 

 much more simply, by common division, arises from the de- 

 velopment of the fraction ; ,v v/ov^^ 

 IV' 1 — A" ,=j1ii?3i3o 

 2(1 — 2Acos6 + A^) ' * • ^i> VJ:rn r iMfJ 



so that, taking account of the remai?ider of the ^ivisibrii'tfte 

 general equivalent of the series is this fraction minus 



cos(n + pe-Acoswg 

 "^ 1 - 2 A cos 9 + A^ • • • • L^-J 



Now confining our attention to the continuous values of A, it 

 is obvious, upon the principles laid down in the former part 

 of this paper, that in the extreme ease of A = 1 and « = 00 , 

 the fraction [2.] vanishes; and [1.] alone correctly represents 

 the sum of the series in the limiting case. 

 ; It may be proper to remark here that [1.] is affirmed to 

 be the true value of the series in the case of A = 1, solely in 

 consequence of that case coming under the control of the ge- 

 neral law of continuity which governs all the cases furnished 

 by the continuous variation of A, cos S being fixed. It is not 

 allowable simultaneously with this continuous variation to in- 

 troduce considerations connected with the variation of cos I?: 



