TABLE VI.-Period of Conjunction of Mercury and Jupiter. (0° denoting Conjunction-63 sets for Kew-43 sets for Trevandrum.) I desire, in conclusion, to thank Mr. William Dodgson, who has given me much assistance in the calculations and diagrams of this paper. III. "Note on the Value of Euler's Constant; likewise on the Values of the Napierian Logarithms of 2, 3, 5, 7, and 10, and of the Modulus of common Logarithms, all carried to 260 places of Decimals." By Professor J. C. ADAMS, M.A., F.R.S. Received December 6, 1877. In the "Proceedings of the Royal Society," vol. xix, pp. 521, 522, Mr. Glaisher has given the values of the logarithms of 2, 3, 5, and 10, and of Euler's constant to 100 places of decimals, in correction of some previous results given by Mr. Shanks. In vol. xx, pp. 28 and 31, Mr. Shanks gives the results of his re-calculation of the above-mentioned logarithms and of the modulus of common logarithms to 205 places, and of Euler's constant to 110 places of decimals. Having calculated the value of 31 Bernoulli's numbers, in addition to the 31 previously known, I was induced to carry the approximation to Euler's constant to a much greater extent than had been before practicable. For this purpose I likewise re-calculated the values of the above-mentioned logarithms, and found the sum of the reciprocals of the first 500 and of the first 1000 integers, all to upwards of 260 places of decimals. I also found two independent relations between the logarithms just mentioned and the logarithm of 7, which furnished a test of the accuracy of the work. On comparing my results with those of Mr. Shanks, I found that the latter were all affected by an error in the 103rd and 104th places of decimals, in consequence of an error in the 104th place in the deter 81 mination of log With this exception, the logarithms given by Mr. Shanks were found to be correct to 202 places of decimals. The error in the determination of log. 10, of course entirely vitiated Mr. Shanks' value of the modulus from the 103rd place onwards. As he gives the complete remainder, however, after the division by his value of log, 10, I was enabled readily to find the correction to be applied to the erroneous value of the modulus. Afterwards I tested the accuracy of the entire work by multiplying the corrected modulus. by my value of log, 10. Mr. Shanks' values of the sum of the reciprocals of the first 500 and of the first 1000 integers, as well as his value of Euler's constant, were found to be incorrect from the 102nd place onwards. Let S,, or S simply, when we are concerned with a given value of n, denote the sum of the harmonic series, Also let R, or R simply, denote the value of the semi-convergent series, where B1, B2, B3, &c., are the successive Bernoulli's numbers. Then if Euler's constant be denoted by E, we shall have and the error committed by stopping at any term in the convergent part of R, will be less than the value of the next term of the series. I have calculated accurately the values of the Bernoulli's numbers as far as B, and approximately as far as B100, retaining a number of significant figures varying from 35 to 20. When n=1000, the employment of the numbers up to B suffices to give the value of R1000 to 265 places of decimals. When n=500, it is necessary to employ the approximate values up to B, in order to determine R500 with an equal degree of exactness. In order to reduce as much as possible the number of quantities which must be added together to find S500 and S1000, I have resolved the reciprocal of every integer up to 1000 into fractions whose denominators are primes or powers of primes. Thus S500 and S1000 may be expressed by means of such fractions, and by adding or subtracting one or more integers, each of these fractions may be reduced to a positive proper fraction, the value of which in decimals may be taken from Gauss' Table, in the second volume of his collected works, or calculated independently. + Thus I have found that: 249 2 3 120 3 86 205 58 1 + + + + 3 + + 11 15 26 32 24 33 27 67 28 38 33 61 45 11 102 68 23 111 $500 = + + + 5 + 137 97 101 103 107 109 113 127 131 25 126 27 28 29 85 88 91 92 + + + + + + + 139 149 151 157 163 167 173 179 181 + + 97 98 100 101 107 113 115 116 118 121 +(the sum of the reciprocals of the primes from 251 to 499) + Similarly I have found that: 249 310 181 75 62 35 220 11 300 726 $1000= + + + + + + + + 729 625 343 121 169 289 361 529 841 512 + 32 34 21 10 40 48 28 56 7 31 40 45 + + + + + + + + + + 961 37 41 43 47 53 59 61 67 71 73 79 25 49 44 69 82 90 104 12 67 84 121 + + + + + + + + + + 83 89 97 101 103 107 109 113 127 131 137 + 85 144 10 26 141 83 34 53 132 171 + + + + + + + + + 163 167 173 179 181 191 211 216- 221 226 47 48 236 49 246 53 + + + + + + + + 251 257 263 269 271 277 281 283 293 307 261 54 266 57 170 175 176 178 181 185 + + + + + + + + + + 311 313 317 331 337 347 349 353 359 367 188 191 193 196 200 202 206 211 212 217 + + + + + + + + + + 373 379 383 389 397 401 409 419 421 431 + 218 221 223 226 230 232 233 235 241 245 + + + + + + + + + 433 439 443 449 4.57 461 463 467 479 487 247 251 + + +(the sum of the reciprocals of the primes from 503 to 997) — 43. This mode of finding S500 and S1000 is attended with the advantage that if an error were made in the calculation of the former of these quantities, it would not affect the latter. The logarithms required have been found in the following manner: log 2=7a-2b+3c, log 3=11a-3b+5c, log 5=16a-4b+7c. Also or again, log 7= (39a-10b+17c-d); log 7=19a-4b+8c+e, and we have the equation of condition, a-2b+c=d+2e, which supplies a sufficient test of the accuracy of the calculations by which a, b, c, d, and e have been found. If we have settled beforehand on the number of decimal places which we wish to retain, and have already formed the decimal values of the reciprocals of the successive integers to the extent required, then the formation of the values of a, b, c, d, and e, will only involve operations which, though numerous, are of extreme simplicity. In this way have been found the following results : Log 109 10536 05156 57826 30122 75009 80839 31279 83061 20372 98327 40725 63939 23369 25840 23240 13454 64887 65695 46213 41207 66027 72591 03705 17148 67351 70132 21767 11456 06836 27564 22686 82765 81669 95879 19464 85052 49713 75112 78720 90836 46753 73554 69033 76623 27864 87959 35883 39553 19538 32230 68063 73738 05700 33668 65 Log 25-2404082 19945 20255 12955 45770 65155 31987 01772 11747 63352 02297 28561 42083 06828 16287 62241 55690 62020 38337 10701 85958 13391 57612 02856 02344 55254 44440 90711 64191 09254 90615 87090 13793 32587 08185 56690 89768 86470 69797 42768 97243 12354 16791 64980 33118 36535 36811 73829 09383 64151 16223 48133 67972 69296 Log 81÷8001242 25199 98557 15331 12931 28631 20890 67623 60339 58145 90685 43409 40510 22236 97287 99924 04408 75833 17607 39941 83907 88915 98331 57135 00593 07313 64880 85644 69078 59065 10006 71375 61155 92285 64823 02773 78467 95356 20673 20672 56121 24774 48623 61600 82118 41837 57253 45313 78157 48027 60627 91715 42041 36587 2 Log 50:49=•02020 27073 17519 44840 80453 01024 19238 78525 33383 73356 83210 27195 49256 65918 71880 87170 92908 14086 00703 48551 55810 69865 22995 29709 68602 61790 51909 27000 19877 96234 68586 52194 37909 61418 83597 32774 05301 16399 74760 65371 30928 59153 97434 74168 79079 46094 49807 56880 62620 29129 95963 65850 08854 45 Log 126÷125-00796 81696 49176 87351 07973 39067 84478 84307 61916 78206 21803 11515 15228 34251 08036 00862 32503 51700 93221 55597 11104 32429 31908 69430 97326 52573 22928 44338 63827 35942 41437 63883 38664 80785 92159 70835 21671 40563 92519 30299 88730 07233 43319 67047 32333 55315 84852 90164 08154 11413 00140 51668 01463 4832 All these are Napierian logarithms. The above-mentioned equation of condition is satisfied to 263 places of decimals. Whence have been deduced the following: Loge 2 = ⚫69314 71805 59945 30941 72321 21458 17656 80755 00134 36025 52541 20680 00949 33936 21969 69471 56058 63326 99641 86875 42001 48102 05706 85733 68552 02357 58130 55703 26707 51635 07596 19307 27570 82837 14351 90307 03862 38916 73471 12335 01153 64497 95523 91204 75172 68157 49320 65155 52473 41395 25882 95045 30081 06850 15 Loge 31.09861 22886 68109 69139 52452 36922 52570 46474 90557 82274 94517 34694 33363 74942 93218 60896 68736 15754 81373 20887 87970 02906 59578 65742 36800 42259 30519 82105 28018 70767 27741 06031 62769 18338 13671 79373 69884 43609 59903 74257 03167 95911 52114 55919 17750 67134 70549 40166 77558 02222 03170 25294 68992 45403 15 Loge 51 60943 79124 34100 37460 07593 33226 18763 95256 01354 26851 77219 12647 89147 41789 87707 65776 46301 33878 09317 96107 99966 30302 17155 62899 72400 52293 24676 19963 36166 17463 70572 75521 79637 49718 32456 53492 85620 23415 25057 27015 51936 00879 77738 97256 88193 54071 27661 54731 22180 95279 48521 29282 13604 17624 80 Loge 71.94591 01490 55313 30510 53527 43443 17972 96370 84729 58186 11884 59390 14993 75798 62752 06926 77876 58498 58787 15269 93061 69420 58511 40911 72375 22576 77786 84314 89580 95163 90077 59078 24468 10427 47833 82259 34900 84673 74412 50497 37048 53551 76783 55774 86240 15102 77418 08868 67107 51412 13480 93879 74210 03537 95 Loge 102 30258 50929 94045 68401 79914 54684 36420 76011 01488 62877 29760 33327 90096 75726 09677 35248 02359 97205 08959 82983 41967 78404 22862 48633 40952 54650 82806 75666 62873 69098 78168 94829 07208 32555 46808 43799 89482 62331 98528 39350 53089 65377 73262 88461 63366 22228 76982 19886 74654 36674 74404 24327 43685 24474 95 |