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XI. Investigation of a New Series fm' the Computation of Logarithms ; mth a New 

 Investigation of a Series for the Rectification of the Circle. By James Thomson, 

 LL.D., Professor of Mathematics in the University of Glasgow. 



Read 7th May 1838. 

 I. 



The series l{\ + x) — M.{x — \a?-\-\a? — \oif^^kc.), discovered by Mebca- 

 TOR, seems to be the origin from which, directly or indirectly, all the series may 

 be derived which are usually employed in the computation of logarithms. A 

 series, which affords remarkable facilities for such computations, and which lately 

 occiirred to me, may be investigated in the following manner. 



In Mercator's series, change x successively into - and — - ; then, by adding 

 Ixio each of the results, we get 



l{x + n)z=lxJrUl __+-__ _ + &c.) (1) 



\x 2 x^ ax' 4 X* / 



^ ' \ X 1x^ 2 x^ \ x^ / ^ ' 



Take half the sum and half the difference of these ; then 



,(,.^.)-,(.-.) =M(^^lg.^lgH...) (1, 



By multiplying the latter by n, and dividing the product by 2 x, we get 



n{li.^n^-lix-n)} /Xn^ 1 .^ 1 .6 x 



\x \1 x' ^ 2.3 a:* ^ 2.5 x^ ^ ) ^' 



Adding this and (3), and by transposition, we obtain 



' "^ - 2 ^ Ix ^ ^^\3.4 X* ^ 5.6 x^ +7.8 xs + '^''■JW 



If w = 1, this becomes 



l{x+l) + l(x-l) l(x+l)-li.-l) /I 12 13 1 .fcA...(7) 



"'- 2 + ix ^ ys.ix*^ 5.6x^^7.8 x^^'' J ^'' 



The m>\ or general term of this series, is evidently M /2m+i)(2m + 2) ' \^/ 



VOL. XIV. PART I. EG 



