Mr. Vince's Supplement , Set. 33 
iir{l a feries, the fame reduction which, from that feries, pro- 
duced the feries §-.i+f~&c. mud alfo have produced 
— $ + 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 § “ 4 + 1 - See. to be divided into two parts i 
the fird part to contain all the terms till we come to thofa 
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 (fuppofmg the fird part to terminate at an even number of 
terms) 1 ~ 1 4 * 1 - 1 + See. fine fine. The fil'd part, by colledb- 
ine; two terms into one* becomes ~ — - — - — — — r-— - — &c*- 
which feries, as it is continued till the terms become infinitely 
fmall, is equal to 1 -f II. L. 2. The fecond part 1 - 1 + 1 - 
&c.has not;, taken abdradtedly of its origin, any determinate value 
(as will be afterwards obferved), but confidered as part of the ori- 
ginal feries it has, for that feries mud have been deduced from the 
expanfion of the binomial 1 , Or —ty ; and hence, when 
#= 1, i - 1 + 1 - Sec. can in this cafe have come only from 
, which, therefore, mud be fubdituted for it ; confe- 
1 + 1 
quently the two parts together give — §4- H. L. 2. 
Having thus explained the nature of the feries which I pro- 
pofed to film, and the principle upon which the corredtioii 
depends, I mud bog leave to acknowledge my obligations to 
my very worthy and ingenious friend George Atwood, Efq. 
F.R.S. who fird obferved that the feries i — 1 + 1 — 1 + &c. has 
no determinate value in the abdradt, as it may be produced by 
— - — — — whatever be the number of units in the denomi- 
1 + 1 + 1 + &c. 
Vol, LXXVo F nator g 
