TRANSACTIONS OF THE SECTIONS. 



17 



Taking (2) in the form 



e=l + 



1+ 6+ 10+..' 



. .., so that ^1 = 1, P2 = ^- 



(3) 



?l=lj ?2=1- 



and writing the convergents^' ^^, . 



the convergents were calculated as far as ^ (which corresponds to the quotient 



150). 



The following Table contains the values of the convergents as far as ^o. 



9'20 



3 



4 



5 



6 



7 



8 



9 



10 



11 



12 



13 



14 



15 



16 



17 



18 



19 



20 



Pn- 



19 



193 



2 721 



49 171 



1 084 483 



28 24.5 729 



848 456 353 



28 875 761 731 



1 098 127 402 131 



46 150 226 651 233 



2 124 008 553 358 849 



106 246 577 894 593 683 



5 739 439 214 861 417 731 



332 993 721 039 856 822 081 



20 651 350 143 685 984 386 753 



1 363 322 103 204 314 826 347 779 



95 453 198 574 445 723 828 731 283 



7 

 71 



1001 



18 089 



398 959 



10 391 023 



312 129 649 



10 622 799 089 



403 978 495 031 



16 977 719 690 391 



781 379 079 653 017 



39 085 931 702 241 241 



2 111421 691 000 680 031 



122 501 544 009 741 683 039 



7 597 207 150 294 985 028 449 



501 538 173 463 478 753 560 673 



35 115 269 349 593 807 734 275 559 



2 599 031 470 043 405 251 089 952 039 



7 064 900 016 612 187 878 152 462 721 

 j»33=5 933 736 817 524 490 649 943 748 885 310 086 922 977 536 976 487 014 058 103 



672 162 883, 

 533=2 182 899 784 489 322 239 844 266 493 459 455 750 162 013 065 305 797 691 300 



833 210 159. 



Since 



+•■ 



we see that e differs from — by less than 



if w=39, 



272. 



rect. 



2 



— =(134 ciphers) 



?» " in1n+l' 'inln+l ^ 



. ; so that 135 figures of the result obtained by dividing 7)39 by ^33 are cor- 

 On performing the division to 137 places and applying the correction for 



-, the value of e was obtained to 137 decimal places, viz. 



inin+i 



6=2-71828 18284 59045 23536 02874 71352 66249 

 77572 47093 69995 95749 66967 62772 40766 

 30353 54759 45713 82178 52516 64274 27466 

 39193 20030 59921 81741 35966 29043 57. .., 



which agrees with Mr. Shanks's calculation obtained from the series 



e=l+l-' 1.1. 



1.2 ' 1.2.3 



1871. 



