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XI. Investigation of a New Series fm- the Computation of Logarithms ; with 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 / (I + «) = M (^ — \o(? + ^a? — i -^^^H- &c.), discovered by Merca- 

 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 

 occurred 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 



/(. + «) = /- + M(__-- + --_--^ + &c.) (1) 



1, N 7 . tit/ ^ 1 «2 1 W^ 1 »* , \ 



/(._«) = Z. + m(----^---3---,-&c.) (2) 



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



n. + .)-^(x-.) ,M(^+lg+lg + .c.) (4) 



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



4^ -^V2^+2:3^ + 2:5^ + ^V ^^^ 



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



^^ = 2 + 4^ + ^(,374 ^ + 576^6+ 778 ^+^"'/^> 



\in= 1, this becomes 



l{x+\)^-l{x-\) l(x+\)-l{x-\) /I 12 13 1 \ 



l^_ + _ + M^^-^ + --+7;g^ + &c.^... (7) 



. m /l\2m + 2 



The mtn, or general term of this series, is evidently M (2m+n(2 — iTV \x) 



VOL. XIV. PART I. E e 



