1879.] the numher of 'primes between limits. 



TABLE VIII. 



307 



The sign + indicates that the values given by the formula 

 are in excess of the true values, and the sign — that they are 

 in defect. 



It thus appears that the numbers derived from (c), even for 

 ranges not exceeding 4,000,000, do not differ from the numbers 

 counted by much more than do those given by Legendre's formula. 

 The extreme errors in the latter case are — O'O and +177'3, the 

 range being 187"2, and in the former case — 16"2 and —226-3, 

 the range being 210-1. 



§ 12. The series used for lia; in § 10 is semi-convergent, and 



all its terms are of the same &ign ; the resulting value of A can 



however be derived from Theorem V, of Tchebycheff's Memoir 



already referred to. This theorem is that if the function which 



denotes the number of primes inferior to x can be represented 



cc 

 algebraically, up to quantities of the order , ' ■ inclusive, by 



means of the functions x, log x, e", it will be represented by the 

 formula 



4- 



log X ' (log xy 



Vol. III. Pt. vii. 



+ 



1 . 2 



{logxy 



+ 



1.2.3... {n-l)a; 



(logo-)" 



23 



