48 Mr Olaisher, Account of an Enumeration [Mar. 12, 



The results of the enumeration are shown in Tables A, B, G 

 (at the end of the paper), which are arranged in an exactly similar 

 manner to Tables I, II, III (pp. 21 — 23) : the explanation of the 

 mode of arrangement is given on p. 19. 



The numbers of primes in each quarter million up to three 

 millions are : — 



FiBST Million. Second Million. Thibd Million. 

 First quarter 22,045 17,971 17,150 



Second „ 19,494 17,682 16,991 



Third „ 18,700 17,455 16,922 



Fourth „ 18,260 17,325 16,822 



Total 78,499 70,433 67,885 



and these may be compared with the similar values for the three 

 millions from 3,000,000 to 6,000,000 on p. 19. 



It will be seen that in the first million 7'8 per cent, of the 

 numbers are primes, in the second million 7'0 per cent., and in the 

 third 6-8, while in each of the seventh, eighth, and ninth millions 

 the percentage is about 6 "3. In the first hundred thousand 

 numbers about 10 per cent, are primes, and this percentage 

 is reduced to 6*7 at the end of the third million, and to 6'3 

 at the end of the ninth million. Throughout the whole of the 

 first million the numbers of primes in each group of 100,000 

 steadily diminish, but this is not the case for the second or third 

 millions, or for any one of the other three millions. 



The numbers of primes in each of the six millions are ; 



NUMBEB OF PeIBIES. DiFFEBENCE. 



First million 78,499 „ .^ „ 



Second „ 70,433 °'^2^ 



Third „ 67,885 ^'^'^^ 



Seventh „ 63,799 ^ . ., 



Eighth „ 63,158 lit. 



Ninth „ 62,760 '^'^^ 



The difference between the numbers of primes in the third and 

 seventh millions is 4,086, giving for each of the four intermediate 

 differences an average of 1021|^. 



Several enumerations have been made of the primes in the 

 first million. In vol. ii. of the third edition of his TMorie des 

 Nomhres (p. 65), Legendre gave the number of primes in the first 

 million as 78,493, which differs from the true value by 6. The 

 following Table contains the results of the enumeration as given 



