522 Mr. J . W. L. Glaisher on the [June 15, 



form a real verification of log 2, log 5, and log 10 ; they also verify P, 

 Q, and K ; but an error in log 3 in the transcription of P, Q, and R, or 

 their multiplication by 11, 8, and 5, or in the final addition and multipli- 

 cation by 2, would not be detected. 



In the above logarithms the last figure may be in error to the extent of 

 one or two units. 



The value of Euler's constant to 100 decimal places is — 

 7 = -57721 56649 01532 86060 65120 

 90082 40243 10421 59335 93992 

 35988 05767 23488 48677 26777 

 66467 09369 47063 29174 67497 . . . 



The last figure here also may be in error to the extent of one or two 

 units. 



It will be observed that Mr. Shanks's value* of y for a?=500 differs 

 from (C) in the 65th decimal place, and that his value for %= 1000 differs 

 from (D) in the 73rd place. This is caused by an inaccurate value of B 13 



nTt . 8553103 



having been made use of. The correct value of B 13 is g ; but Euler, 



who first calculated it, made it ('Acta Petropolitana ' for 1781, 



p. 46), and this incorrect value is given in the ( Penny Cyclopsedia ' 

 (Article f< Numbers of Bernoulli") and probably in other places. 



The values of the first thirty-one Bernoulli's numbers are given in a 

 paper by Ohm (Crelle's Journal, t. xx. p. 11), and B 13 is given correctly 

 there. The agreement of the values of Euler's constant contained in this 

 paper (when the logarithms of 2 and 5 are corrected) afford a complete 

 verification of the Bernoulli's numbers as far as B 2(? , and partial verifica- 

 tions of the rest. 



The difficulty and inconvenience of making calculations to so many de- 

 cimal places is sufficient to warrant the publication of the values of the 

 positive and negative parts of the portion of the series involving the Ber- 

 noulli's numbers, in case any one should desire to repeat any part of the 

 calculation. We have 



11 1 _ 1 B B B 



+ T + T • + ^- l0 S^-2^ + 2^-4P + 67---* ; 



and if m denote the sum of the terms of the same sign as the harmonic 

 series, and n the sum of the terms of the same sign as the logarithms, 

 viz. if 



__! ?? A 



n ~2^ + 4^ + 8*„ + 



* Proc. Koy. Soc. vol. xv. p. 432. 



