298 Mr Glaisher, On Legendre's formula for [Dec. 8, 



unaltered, but the two tables (viz. from to 400,000 and from 

 400,000 to 1,000,000) are united in a single table*. 



§ 2. In the letter written by Gauss to Encke on December 

 24, 1849 {Werke,t. ii. pp. 444 — 447), reference is made to the value 

 of the constant in Legendre's formula. An account of a portion of 

 this letter and of the enumeration which accompanied it, is given 

 ante, pp. 49 — 50, 



Having, with the assistance of Goldschmidt, made an enumera- 

 tion of the primes up to 3,000,000, Gauss compares the numbers 

 for each half-million with the corresponding values of the loga- 

 rithm-integral li X, and then refers to Legendre's formula, which, 

 he states, he had either overlooked or forgotten till Encke drew 

 his attention to it. Gauss then gives the results of a comparison 

 with the values obtained by Legendre's formula, and remarks that 

 although these differences are less than in the case of the liic 

 formula, they seem to increase more rapidly; he proceeds "Um 

 Zahlung und Formel in Uebereinstimmung zu biingen, miisste 

 man respective anstatt A = 1 •08366 setzen 



1-09040 

 1-07682 

 1-07582 

 1-07529 

 1-07179 

 1-07297 



" Es scheint, dass bei wachsendem n der (Durchschnitts-)Werth 



von A abnimmt, ob aber die Grenze beim Wachsen des n ins Un- 



endliche 1 oder eine von 1 verschiedene Grosse sein wird, dariiber 



wage ich keine Vermuthung." 



Professor Tchebycheff subsequently proved in his memoir "Sur 



la fonction qui determine la totalite des nombres premiers in- 



ferieurs a une limite donnde"-]-, that \if{x) denotes the number of 



cc 

 primes inferior to x, then, if jj—^ — log x has a limit, it must be 



* On page 49 of this volume of Proceedings I have given a hst of errata in this 

 table. On comparing it however with the two tables in the second edition and its 

 supplement, I find that one of the errata is due to a misprint ; in the second edition 

 the number of primes counted up to 300,000 is given as 25,998, and this number is 

 misprinted 25,988 in the third edition. But for this misprint the number of primes 

 between 250,000 and 300,000 would appear as 3953 which is correct, and the 

 number between 300,000 and 350,000 as 3979 which is in error by only a unit. 

 The two errata of 10 and 9 in these groups, noted on page 49, are therefore due to 

 a real error of 1, and the misprint of an 8 for a 9. 



t Mem. de VAcad. de St Petershourg (Savans Etrangers), t. vi. (1851), pp. 141 — 

 157: reprinted in Liouville's Journal, t. xvii. (1852), pp. 341 — 365. 



