1871.] Numerical Value of Euler's Constant. 31 



distinct sets of numbers, increased by tbe sum of the series to 5000 terms, 

 will give the sum of the harmonic series to 10,000 terms. 



Before proceeding further it should be stated that, having obtained the 

 correct value of E, from S 200 &c. to 110 decimals, verifying Mr. Glaisher's 

 to 99 decimals, it was comparatively easy to extend S 500 and S 1000 to 110 

 decimals, and to correct and extend S 2000 , S 5000 , and S 10>000 to the same extent. 



When we have S 100 , the calculations from Bernoulli's 31 numbers will 

 lead to obtaining E only to about 92 decimals. This value may no doubt 

 be extended by finding the ratio between the last and each succeeding 

 Bernoulli's number. Such ratio is, however, only approximative, and can 

 yield correct results of only a limited number of decimals. The excess of 

 the + Bernoulli terms over the — ones, to 110 decinals, when S 100 is used, 

 is readily obtained when E and log e 100 are known to the same extent. 

 Such excess will be found below ; also the separate sums of the + and — 

 terms in which Bernoulli's numbers enter, both when S=100 and when 

 S=200, to 205 decimals. 



The values of S 500 , S 100o , S 2000 , S 5000 , and S 10)000 to 110 decimals, also the 

 corresponding -+- and — results of the Bernoulli terms to the same extent, 

 are likewise given below, as they involve very considerable calculation, and 

 may thus be tested and verified. The values of S 100 and S 200 may as well 

 be also written anew, inasmuch as a few slight errors had crept into them 

 before. 



E= '57721 



56649 



01532 



86060 65120 



90082 



40243 



1 042 1 



59335 93992 



35988 



05767 



23488 



48677 26777 



66467 



09369 



47063 



29174 67495 



11141 



1 442 1. 















S100 =5*18737 



75176 



39620 



26080 51176 



75658 



25315 



79089 



72126 70845 



16531 



76533 



95658 



72195 57532 



55049 



66056 



87768 



92312 04135 



49921 



06986 



97779 



79182 73403 



18717 



00828 



94825 



42444 49096 



57618 



56474 



16326 



13467 07313 



21 1 14 



47132 



49733 



09103 51129 





09481 



21444 



47605 73863 



97130 



86163 



68374 



00246 53024 



30844 



64971 



9447^ 



28783 30029 



84018 



r 5499 



64301 



86679 89238 



37326 



83211 



85439 



0591 1 76542 



77755 



27568 



86559 



30203 06046 



25715 



75389 



22254 



75748 47845 



75246 



64079 



54805 



61627 08880 



S500 =679282 



34299 



90524 



60298 92871 



45367 



97369 



48198 



13814 39677 



91166 



43088 



89685 



43566 23790 



55049 



24576 



49403 



73586 56039 



14705 



68279, 















S iooo =7-48547 



08605 



50344 



91265 65182 



04333 



90017 



65216 



79169 70880 



36657 



73626 



74995 



76993 49 l6 5 



20244 



°9599 



34437 



41184 50813 



93907 



7**34- 















S 2000 =8-17836 



81036 



10282 



40957 76565 



71641 



69368 



79354 



66740 91248 



77402 



20419 



74812 



15302 80688 



34328 



60377 



35324 



29687 02614 



20643 

















