TUANSACTIONS Or THE SECTIONS. 



15 



Having computed the values of 31 of Bernoulli's numbers in addition to those 

 previously known, I have been able to find a much more approximate value of E 

 than has been before obtained. 



In fact, by putting n = 1000 in the above formula, the value of E is correctly found 

 to more than 260 places of decimals. 



In order to diminish as much as possible the number of decimal fractions which 

 mii3t be added together to form S 1000 , 1 first resolved the reciprocal of every integer 

 up to 1000 into ordinary fractions whose denominators were primes or powers of 

 primes, and then combined the fractions corresponding to each of these primes into 

 one sum, so that finally the number of decimal fractions to be added together was 

 reduced to the number of prime numbers below 1000. 



Also the value of log 1000 or 3 log 10 was derived from the Napierian loga- 



rithms of 



10 25 



81 



9 



, and 575, which may be found by extremely simple operations 



when the reciprocals of the successive integers are supposed to have been pre- 

 viously expressed in repeating decimals. 



The following Table gives the results which I have thus obtained : — 



B 



1000 



3 1000 



Log 1000 



•ooooo 

 886n 



4373 6 

 33894 



3i54 6 

 86978 



7-48547 



3 66 57 

 96798 

 92966 

 87809 

 21027 



6-90775 

 89280 

 25903 



345° 6 

 59268 

 23212 



00833 

 32124 

 78499 

 07843 

 74821 

 52223 



08605 

 73626 

 01438 

 39*78 

 5 6 Si5 

 95267 



52789 

 99983 

 35212 

 84487 

 96133 

 72982 



33325 

 18782 

 44114 

 88131 

 27649 

 81562 



ooooo 



98862 



24665 



36054 

 54293 



50344 91265 



74995 76993 



22544 03715 



33827 35905 



86955 67800 

 06455 



39682 

 06644 



374 2 3 

 55889 

 18527 



65182 

 49165 

 81484 



57913 

 24804 



49801 

 51967 

 82138 

 69002 

 10448 



59487 

 06850 



5° 2 59 

 08034 

 88349 



73^37 

 04241 

 70190 

 44545 

 55931 



84632 

 14869 

 89962 

 27898 

 43201 



1 1 743 

 65631 

 61572 

 47738 

 82238 



82137 05205 39743 



70290 27178 29032 



68587 45900 22857 



21624 97666 40425 



19788 65384 90098 

 31056 



04333 90017 65216 79169 70880 



20244 09599 34437 41184 50813 



21958 84703 40431 40398 43368 



00071 54692 68403 25933 79804 



71415 08712 32350 00711 42865 



64053 09262 28033 04465 88631 



05744 07079 91615 26879 4895° 



63952 48420 26999 88621 07296 



31399 68447 86995 95585 18051 



66686 30946 59660 23963 10024 



E '57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 



35988 05767 23488 48677 26777 66467 09369 47063 29174 67495 



14631 44724 98070 82480 96050 40144 86542 83622 41739 97644 



92353 62535 00333 74293 73377 37673 94279 25952 58247 09491 



60087 35203 94816 56708 53233 15177 66115 2S621 19950 15079 



84793 745° s 56961 



The figures in the last two decimal places in each of these quantities are uncertain. 



On a Simple Proof of Lambert's Theorem. 

 Bij Professor J. C. Adams, M.A., F.R.S. 



The following proof of Lambert's Theorem, which I find among my old papers, 

 appears to be as simple and direct as can be desired. 



Let a denote the semiaxis major and e the excentricity of an elliptic orbit, n the 

 mean motion, and /x the absolute force. 



Also let r, r' denote the radii vectores, and u, u' the excentric anomalies at the 

 extremities of any arc, k the chord, and t the time of describing the arc. 



Then 



r = «(1 -e cos m), r' = a(l - e cos *«'), 



k-=a\coau- cosw') 2 +« 2 (l- e2 )( sinM - sin?*') 2 , 



