1879.] the number of primes between limits. SOo 



going table and of Table V. the values of the hyperbolic logarithms 

 used were correct to seven decimal places. 



§ 10. Gauss, Hargreave *, and Tchebycheff "f- obtained as a 

 formula for the number of primes inferior to x the logarithm- 

 integral Mx ; and for this to agree with 



log X — A^ 



the value of ^ as a function of x must be determined by the 

 equation 



that is 



A = log X — ,-. — . 

 ^ \\x 



Replacing li x by its expansion in a semi-convergent series, viz. 

 X f^ 1 2! 8! 



il ^ = , 1 + F Y TT, + -j Tj + &C. 



log X \ log X (log X) (log x) 

 we have 



X ,/^ 1 2 6 „ 



where I denotes log x, whence 



^ _7_1 _l_?_l^_?i ^ 

 \ix~ ^ I V V I' ■' 



and therefore 



. , 1 3 13 71 p 



A = l + 1 + 7, r^ + y] Ts + n ^ + ^^^ 



logx (logo;) (log a;) (log.'r) 



As a first approximation we have 



log X ' 

 and the formula becomes 



l02f X — 1 — :. 



° locr a; 



• Philosophical Magazine, t. xxxv. (1849), pp. 36—53, and t. viii. (1854), 

 pp. 114—122. 



t In the memoir cited in the note on p. 298: the formula, more exactly, is 

 H a; - li 2 ; see § 12. 



