1871.] On the Numerical Value of Euler's Constant. 



29 



The remainder is as follows : — 



17550 06500 84834 29272 80492 56652 30056 77179 51985 96380 

 06400 15769 36917 76044 90943 15598 00090 70477 96549 03362 

 30614 39569 71063 83855 24053 35869 04219 87709 24604 49236 

 79071 67965 18350 79865 61803 71534 89641 16619 31638 84825 

 79008 o 

 Aug. 28th, 1871. 



II. Second Paper " On the Numerical Value of Euler's Constant,, 

 and on the Summation of the Harmonic Series employed in ob- 

 taining such V alue." By William Shanks, Houghton-le-Spring, 

 Durham. Commuuicated by Prof. G. G. Stokes, Sec. U.S. Re- 

 ceived August 30, 1871. 



Three cases and sources of inaccuracy in finding the value of E in the 

 former paper (Proc. Roy. Soc. vol. xv. p. 429) having been pointed out 

 by Mr. Glaisher, and some other minor errors not noticed by him having 

 since been detected by the author, and these having vitiated the results, 

 but only in a slight degree the inferences drawn from them (for in the 

 former paper the last and leading conclusion as to the value of E, though 

 limited, was certainly correct), the author has been led, from a deep sense 

 of obligation to the Royal Society, to revise, correct, and extend what he 

 had previously done. And it will be seen, from comparing Mr. Glaisher's 

 remarks and results with what follows in this paper, that the supplementary 

 matter herein given, including the extension of E &c. to 110 places of 

 decimals, can scarcely be without interest to mathematicians, and especially 

 as regards the summation of the harmonic series in the formula for finding 

 the value of E. 



Not having seenM. Oettinger's article in Crelle's c Journal,' "On Com- 

 puting the value of E," the author is unable to state what artifices he used 

 in summing the harmonic series. Mr. Glaisher gives a very simple and 

 obvious one from M. Oettinger, which the author could not but see and 

 employ for calculating the values of the reciprocals of the even numbers. 



In summing the harmonic series, the author found the reciprocals of all 

 numbers up to 200, as far as 200 places of decimals ; next the reciprocals 

 from 200 to 5C0, to only 105 decimals; and afterwards the reciprocals of 

 the odd composite numbers up to 5000, to the same extent. In passing 

 from S 2000 to S 5000 some extra calculation was necessary, which need not be 

 stated here. It is, however, necessary to calculate, in extenso, the reci- 

 procals of the odd composite numbers only to half the number of terms 

 which it is proposed to sum. The reciprocals of all the prime numbers 

 must of course be calculated separately. 



The leading artifices the author employed to shorten calculation may be 

 best stated and explained by supposing that the reciprocals of all the odd 

 numbers below 5000 have been computed and retained separately, also 



