28 



Mr. W. Shanks on Logarithms, §c. 



[Nov. 23, 



that of M, are given. Quite recently Mr. Glaisher has shown that the values 

 of these logs are inaccurate from about the 59th decimal. This inaccuracy 

 the author regrets, and has therefore recalculated them, and now gives 

 their values entire and corrected. The values of logs e 2, 3, 5, and 10 agree 

 with and verify Mr. Glaisher' s, as will be seen, to about the 100th decimal ; 

 and it is thought (from this and the great care used) that the remaining 

 decimals now given are free from error. The value of e, given in the 

 former paper, has lately been verified by Mr. Glaisher to 137 decimals. 

 The value of M has also been recalculated ; and the accuracy of its value 

 (as far as division is concerned) may be relied on, inasmuch as proofs 

 were taken, by casting out the nines, at every interval of five quotient 

 figures. This number, if 205 ciphers be prefixed to it and the decimal 

 point be removed and placed on the left of them, may be regarded as the 

 longest non -circulating reciprocal on record, consisting of 410 decimals. 

 The remainder after the last division is given, so that the accuracy of the 

 division may thereby be easily tested, by contraction or otherwise. 

 The corrected values are as follow : — 



Lo ge 2 = -69314 71805 59945 30941 



52541 20680 00949 33936 



42061 48102 05706 85733 



07596 19307 27570 82837 

 01018+&C. 



Log e 3 = i"C986i 22886 68109 69139 



945 J 7 34 6 94 333 6 3 74942- 



88070 02906 59578 65742 



27741 06031 62769 18338 

 • 02948 + &c. 



Log e 5 = 1-60943 79124 34100 37460 



77219 12647 89147 4*7^9 



00106 30302 17155 62899 



70572 75521 79637 49718 

 51474+&C 



Log e 10 = 2*30258 50929 94045 68401 



29760 33327 90096 75726 



42167 78404 22862 48633 



78168 94829 07208 32555 

 52492+&C. 



M= '43429 44819 03251 82765 



65661 14453 78316 58646 



18668 38440 53639 80947 



56816 80637 45310 65045 

 73760 + &C. 



72321 21458 17656 80755 00134 36025 



21969 69471 56058 63326 99641 86875 



68552 02357 58130 55703 26707 51635 



14351 90307 03862 38916 73471 12335 



52452 36922 52570 46474 90557 82274 



93218 60896 68736 15754 81373 20887 



36800 42259 30519 82105 28018 70767 



13671 79373 69884 43609 59903 74257 



07593 33226 18763 95256 01354 2685! 



87707 65776 46301 33878 09317 96108 



72400 52293 24676 19963 36166 17463 



32456 53492 85620 23415 25057 27015 



79914 54684 36420 7601 1 01488 62877 



09677 35248 02359 97205 0S959 82983 



40952 54650 82806 75666 62873 69098 



46S08 43799 89482 62331 98528 39350 



11289 18916 60508 22943 97005 80366 



49208 87077 47292 24949 33843 17483 



79768 71211 65951 73183 60409 55627 



32572 68778 26750 65871 94503 27872 



