308 Mr Qlaisher, On Legendre's formula for [Dec. 8, 



The use of tbis theorem amounts to that of the semi-con- 

 vergent series in § 10. 



The logarithm-integral li x is defined by the equation 



,. r* di 

 11 



/: 



' log ^ ' 



and therefore contains an infinite element corresponding to w = 1 ; 

 this element however is supposed to be omitted and the principal 

 value taken. Hargreave's formula for the number of primes in- 

 ferior to X was li X, but Tchebycheff gave 



du 



' 2 log U ' 



that is li cc — li 2, in which the infinite element is not included 

 between the limits of the integration ; the value of li 2 is however 

 only 1'04516 ..., so that the difference between the formulse is 

 unimportant even for comparatively small values of x. 



As already mentioned Gauss excluded unity in his enumera- 

 tion; this appears from the number of primes inferior to 1000 

 which he gives as 168 *. Legendre however included unity, for 

 he gives the number of primes inferior to 10,000 as 1230; and I 

 have followed him in including it in the values of ^ {x) given in 

 this paper. 



§ 13. With regard to the manner in which Legendre obtained 

 his constant 1'08366, it does not seem to have been determined 

 solely from the value 07=10,000; for, if this were the case, we 

 should have 



^ — log 03 — J230 



= 1-08026, 



which differs from 1*08366 by 000340. Legendre's formula for 

 a; = 10,000 gives 1230*51. The constant does not appear to have 

 been determined from any single value of x ; and it seems likely 

 that it was so chosen as to represent as nearly as possible the 

 results of the earlier enumerations. When Legendre subsequently 

 obtained the enumeration for the numbers from 400,000 to 

 1,000,000 the values given by the formula agreed so well with 

 the numbers of primes counted as to apparently confirm the value 

 which had been assigned to the constant; and he had therefore 

 no inducement to examine the question further. 



In this paper I have purposely omitted any comparisons with 

 the lio; formula, as it seems desirable to defer these until the 



* See p. 51 ante. In line 7 of tbis page "has not counted 1 and 2 as primes" 

 should be "has not counted both 1 and 2 as primes", as is clear from the 

 context. 



