1871.] 



Calculation of Eider's Constant. 



517 



when m is large, 



agreeing with (iii). 



The following Table, to replace that in vol. xv. p. 432, was calculated in 

 the way indicated above. 



Number of decimals 



x. Least term. directly obtainable 



from the formula. 



3rd 2 



7th 5 



16 th 13 



32nd 27 



63rd 54 



I57th 136 



314th 273 



629th 546 



1571st 1365 



3142nd 2729 



1 



2 

 5 

 10 



20 

 50 

 100 

 200 

 500 

 1000 



The number of decimal places practically obtainable is limited by the 

 difficulty of calculating the Bernoulli's numbers, of which only thirty-one 

 have been hitherto obtained. By means of these, however, 156 decimals 

 could be obtained of y when #=100 ; and by deducing a few of the subse- 

 quent terms, each from its predecessor (knowing their ratio), 20 places 

 more could be obtained without difficulty. 



It is clear therefore that Mr. Shanks's values of y obtained from #=500 

 and #=1000 ought to agree beyond the 59th decimal, if correctly calcu- 

 lated. The author at first supposed that the want of agreement was due 

 to an insufficient number of the terms involving the Bernoulli's numbers 

 having been included, and he therefore undertook the calculation of this 

 portion of the expression for # = 500 and #= 1000 to 100 decimal places ; 

 the results, however, still showed a difference in the 59th place. 



In order to determine where the error existed, the same portion of the 

 constant was calculated also to 100 places, both from #=100 and #=200 ; 

 all the calculations were performed wholly in duplicate, and so much care 



was taken that the author felt a strong conviction of thei 



r accuracy. 



it 



should be noticed that the agreement of all the four results would not 

 necessarily prove the accuracy of that value of y ; for any error made in 



\ . . , . would merely pro 



2 100 * * 



the calculation of the harmonic series 1 



