VOL. 



LXXV.] PHILOSOPHICAL TRANSACTIONS. 639 



whose value I want, is expressed by the above series, and which arose from the 

 necessity of expanding some quantity in the preceding part of the operation, 

 surely no one can deny that I may substitute for it — -L + h. l. 2. For what- 

 ever quantity it was, which by its expansion produced at first a series, the same 

 reduction which, from that series, produced the series i — •§- + f — &c. must 

 also have produced — 4- + h. l. 2, from the quantity which was expanded. This 

 value of the series I obtained in the following manner. I supposed the series 

 4. — •§-+ -f- — &c. to be divided into 2 parts; the first part to contain all the 

 terms till we come to those where the numerators and denominators become both 

 infinitely great, in which case every term afterwards may be supposed to be equal 

 to unity; the 2d part therefore would necessarily be, supposing the first part to 

 terminate at an even number of terms, 1 — ] + 1 — 1 + &c. sine fine. The 

 first part, by collecting 2 terms into 1, becomes — — — — - — - — ^-_- — &c. 

 which series, as it is continued till the terms become infinitely small, is equal to 

 — 1 + h.l. 2. The 2d part 1 — 1 + 1 — &c. has not, taken abstractedly of 

 its origin, any determinate value, as will be afterwards observed, but considered 

 as part of the original series it has, for that series must have been deduced from 

 the expansion of the binomial (1 + #) - ', or -— — ; and hence, when x = 1, 



1 — 1 + 1 — &c. can in this case have come only from — -, which, therefore, 

 must be substituted for it; consequently the 2 parts together give — -l + h. l. 2. 

 Having thus explained the nature of the series which I proposed to sum, and 

 the principle on which the correction depends, I must beg leave to acknowledge 

 my obligations to my very worthy and ingenious friend George Atwood, Esq., 

 f. r. s., who first observed that the series 1 — 1 + 1 — 1+ &c. has no deter- 

 minate value in the abstract, as it may be produced by 1 &c whatever be 

 the number of units in the denominator;* and it may also be added, that the 

 same series arises from * , , , , C . > provided the number of units be 

 greater in the denominator than in the numerator. The correction will there- 

 fore be different in different circumstances, and will depend on the nature of the 

 quantity which was at first expanded. In the 3d part of my paper, I applied the 

 correction to those cases where the original series arose from the expansion of a 

 binomial, where the correction is in general as I there gave it; but as I did not 

 apply my method to any other series, I confess that it did not appear to me, that 

 the correction would then be different, which it necessarily would, had I ex- 

 tended my reasoning to other cases. I shall therefore add one example to show 



* I have been since informed by Mr. Wales, f. r. s., thata pupil of his, Mr. Bond, made the 

 same observation. — Orig. 



