Mr, Vince's Supplement, See. 33 

 firfl a feries, the fame redudion which, from that feries, pro- 

 duced the feries |-i--{-|-&c. muft alfo have produced 

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

 value of the feries I obtained in the following manner, i fup- 

 pofed the feries | — .1 -f-f - &c. to be divided into two parts ; 

 the firft part to contain all the terms till we come to thofe 

 where the numerators and denominators become both infinitely" 

 great, in which cafe every term afterwards m.ay be fuppofed to 

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

 be (fuppofmg the firft part to terminate at an even number of 

 terms) i - i + i - 1 -\- ^z. fine fine , The firfl: part, by collect- 

 ing: two terms into one, becomes - — ~- &;c. 



^ 2 .3 4- S 6 • 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 abftradedly 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 muft have been deduced from the 



expanfion of the binomial i + x\ , or 7—; and hence, when 



x—\y I - I + I - &c. can in this cafe have come only from 



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

 I + 1 



quently the two parts together give — | + H. L. 2. 



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

 jDofed to fum, and the principle upon which the corredlioii 

 depends, I muft beg leave to acknowledge my obligations to 

 my very worthy and ingenious friend George ArwooD-Efq. 

 F.R.S. who firfl obferved that the feries i - i + i - i -f &c. has 

 no determinate value in the abfira6t, as it may be produced^ by 



^ r— whatever be the numiber of units in the denomi- 



I + I + I + (N-C. 



Vol. LXXV. F iiator^ 



