of the Sums of Neutral Series. 139 



Postscript. 



I have just read Prof. Young's second paper, and without 

 entering into all the windings of his argument, I shall proceed 

 to animadvert upon such parts of it as refer to my own. 



After some preliminary matter the Professor proceeds, 

 "Let us now examine the series 



— + A cos I + A 2 cos 2 + &c. + A" cos n 0, 



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 



l=^ . . . . ri.i 



2(1 -2Acos0 + A 2 )' L J 



so that, taking account of the remainder of the division, the 

 general equivalent of the series is this fraction minus 



A „4.i cos (rc + 1)0 -Acqs w0 f2l 



l-2Acos0 + A 2 * * ' ' 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 case of A = 1 and n = oo , 

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

 the sum of the series in the limiting case." 



What is meant by " confining our attention to the conti- 

 nuous values of A?" Can we draw any conclusion from the 

 equation 



— + A cos + A 2 cos 2 + &c. + A ra cos n 0, 



_ 1— A 2 , cos(w + l)0— A CO S 72 



~2(l-2Acos0 + A 2 ) 1 — 2Acos0 + A 2 



other than the following, viz. that so long as A is positive, 

 and differs from 1 by an actual magnitude, in the limit when 



n ■=. oo , 



— + A cos + A 2 cos 2 + &c. in inf. 



£i ... 



1 -A 2 



2 (1 - 2 A cos + A 2 ) ' 

 and that when A ceases to differ from 1 by an actual magni- 

 tude, the same series 



- f 1— A 2 T f CQS(w+l)0 — COSM0 \ 7 



'' l2(l-2Acos0 + A 2 )J A=1 L 2(l-cos0) J n=w * 



L2 



