1879.] 



the nu7iiber of primes between limits. 



299 



unity; from which it follows that in the formula 



log X — A^ 

 as X approaches infinity, the limiting value of A must be unity. 



§ 3. The enumerations employed by Gauss were very in- 

 accurate as appears from the lists of errors on pp. 51 — 52 of the 

 present volume, and it seemed desirable to obtain the correspond- 

 ing values of A, using the results of the enumeration of the first 

 three millions given on p. 48. The following table shows the values 

 of ^{x), the actual number of primes inferior to ic, and also the 

 values used by Gauss: — 



TABLE I. 



Taking the values in the {x) column, the corresponding 

 values of A {i.e. the values of A which are such that the number 

 given by the formula is equal to the number of primes counted, 

 for the particular value of x) are shown in the second column of 

 the next table, the third column of which contains Gauss's values 

 of A, and the fourth the differences. 



TABLE II. 



* In the enumerations in this column unity has been counted as a prime ; in 

 Gauss's enumerations, given in the next column, it has not been included. See 

 § 12. 



