Mr. Vince's Supplement^ &c. 3^ 



firfl a ferles, the fame redu«5lIon which, from that feiies, pro- 

 duced the feries i-*-+|— &c. mull: alfo have produced 

 — I + H. L. 2. from the quantity which was expanded. This 

 value of the feries I obtained in the following manner. I fup- 

 pofed the feries I — 4 + ? "" ^^' ^^ ^^ divided into two parts ; 

 the firft part to contain all the terms till wc come to thofc 

 where the numerators and denominators become both infinitely 

 great, in which cafe every term afterwards may be fuppofed to 

 be equal to unity : the fecond part, therefore, would neceflarily 

 be (fuppofing the firll: part to terminate at an even number of 

 terms) i - i + i - i ■{■Scz.fnefme, The firft part, hy coUed- 



inp; two terms into one, becomes ^- &c. 



^ 2.34-56.7 



which feries, as it is continued till the terms become infinitely 

 fmall, is equal to - i + H. L. 2. The fecond part i - i + i — 

 &c.has not, taken abftra6tedly of its origin, any determinate value 

 (as will be afterwards obferved), butconfidered as part of the ori- 

 ginal feries it has, for that feries muft have been deduced from the 



expanfion of the binomial i +x\ , or — — ; and hence, when 



J?= I, I - I + I -&c. can in this cafe have come only from 



, which, therefore, mull: be fubftituted for it ; confe- 



quently the two parts together give — f -}- H. L. 2. 



Having thus explained the nature of the feries which I pro- 

 pofed to fum, and the principle upon which the corre6lioit 

 depends, I muft beg leave to acknowledge my obligations to 

 my very worthy and ingenious friend George ATwooD,Efq. 

 F.R.S. who firft obferved that the feries i — i + i — i + &:c. has 

 no determinate value in the abftradb, as it may be produced by 



— 1 — ; — r-; — whatever be the number of units ia the denomi- 



J + I + I +&C. 



Vol. LXXV. F nator; 



