154 



Mr. W. Shanks on Eider's Constant, 



[itecess^ 



IV. SecoDd Supplementary Paper on the Calculation of the Nume- 

 rical Value of Euler's Constant/' By William Shanks, 

 Houghton-Ie- Spring, Durham, Communicated by the Rev. 

 Professor Price, F.R.S. Received August 29, 1867. 



When n=2m0, we have 



^8-17836 81036 10282 40957 76565 71641 69368 79354 

 66740 91251 77402 20409 26320 14205 58039 78429 

 87946 27554 87631 13645 + 

 £ = •57721 56649 01532 86060 65120 90082 40243 10421 59335 

 93995 35988 05772 51046 48794 94723 80546 (last term is 







24 . 2000'^^' 



Here the 60th decimal place in the value of E is the same when n is 2000 

 as it is when n is 1000. 



When ;i = 500, we have in the value of E, 60th 1 -, r-ooc- aoch'- o 



decimal and onwards J ^ ^^^^^ 



1000, „ „ „ 2 02455 61942 &c. 



2000, » „ 2 51046 48794 &c. 



By subtracting the first of these three from the 1 .o-nn ^oc^c- 

 J ® 1 > 48d90 1326a 



second, we nave j 



By subtracting the second from the third, we have 48590 86852 



It is somewhat remarkable that these differences are the same to five 

 places of decimals ; and it may be observed that the value of E vsdll pro- 

 bably be changed and extended very slowly indeed by employing higher 

 values of n. The remark in the previous Supplementary Paper*, as to n 

 being 50000 or even 100000 in order to obtain probably about 100 places 

 of decimals in E, seems, the author now thinks, to be not well founded ; 

 and he hesitates even to conjecture what number of terras of the Harmonic 

 Progression should be summed " to ensure accuracy in the value of E to 

 100 decimals. 



* Proceedings, vol. xv. p. 429. 



