1871.] 



Calculation of Euler's Constant. 



519 



From x = 200 :— 



y = -57721 56649 01532 86060 65120 



90082 40243 10421 59335 93995 



35988 05770 91390 77706 70283 



52607 02838 27101 38332 64367 



(B). 



From X = 500 :— - 



y = -57721 

 90082 

 35988 

 20064 



56649 

 40243 

 05771 



01532 

 10421 



53864 



86060 

 59335 

 75089 



75314 10504 07812 



65120 

 93995 

 05973 

 29873 



(C). 



From x= 1000 



•57721 

 90082 

 35988 

 50309 



56649 

 40243 

 05772 

 65017 



01532 

 10421 

 02455 

 53150 



86060 

 59335 

 61942 

 61852 



65120 

 93995 

 00398 

 03044 . . 



■ (I>). 



The terms involving B 30 , B 3l , B 22 , and B 19 were the highest used in the 

 calculation of (A), (B), (C), and (D) respectively. 



It will be seen that (A), (B), (C), and (D) differ in two respects : (i) the 

 50th figure in (A) is 2, while in (B), (C), and (D) it is 5, and (ii) all four 

 values are totally different after the 59th figure. 



The first discrepancy (i) pointed to an error in the summation of the 

 harmonic series. As the author had verified Mr. Shanks's value of 



100 : 



it was practically certain that (A) was the correct value. 



To place this beyond all doubt, however, the author calculated y from x 

 = 50 to 57 places of decimals, and the result entirely confirmed (A), it 

 follows, therefore, that Mr. Shanks has made an error of 3 in the 50th 

 1 1 1 



place in the calculation 



100+101 



200' 



50 that we have the fol- 



lowing errata (vol. xv. p. 431) : — In 1 + ^- . . . +^y> the tenth group of 



five should be . . . 53024 . . . instead of . . . 53027 . . . 

 1 1 



In 1 . . . +^Jq the tenth group should be . 39677 . . . instead 



2 ' 



of . . . 39680 

 Inl+y. 



'0880 . . . instead 



. . + iquq tlac tenth group should be . 



of . . . 70883 . . . 



With regard to the second error (ii), which causes the disagreement of 

 all the figures after the 59th, it is clear that no error in the Bernoulli's 

 numbers could produce such a discrepancy, as the terms involving the 

 latter vary their position in each calculation, so that an error in any one 

 of them could not affect the same decimal in (A), (B), (C), and (D). 



