32 



REPORT — 1880. 



The sixth million is still in tie press, and the pi'inbing and stereotyping 

 of the table will be completed early next year. It is intended to prefix 

 to this volume an introduction containing the results of the enumerations 

 for the whole nine millions over which the printed tables will then extend, 

 with comparisons of the numbers found by counting with those given by 

 Legendre's formula and the li x formula. A table of the values of li x 

 from fB = to a; = 9,000,000 at intervals of 60,000 is now in course of 

 calculation, as also is a table of the values given by Legendre's formula 

 for the same arguments. The results of these comparisons for intervals 

 of 250,000 up to 5,000,000 are given in Section IV. 



II. Results of the Enumeration of the Primes in the Fifth Million. 



The following table, which is similar to that given on p. 47 of last 

 year's Report, contains the chief results of the enumeration of the primes 

 in the fifth million, arranged according to the numbers of primes in 

 the centuries. 



4,000,000 to 5,000,000. 



This table shows the number of centuries in each group of 100,000, 

 each of which contains no prime, each of which contains one prime, 

 two primes, &c. For example, between 4,000,000 and 100,000 there is 

 no century containing no prime {i.e. consisting wholly of composite 

 numbers), there are three centuries which contain each one prime, fifteen 

 which contain two primes, and so on, there being only one which contains 

 fourteen primes. The number at the foot of each column is the total 

 number of primes iu the group of nnmbers to which the column relates ; 

 thus, for example, there are 6,628 primes between 4,000,000 and 

 4,100,000. 



The next table shows the numbers of primes in each successive group 

 of 10,000 between 4,000,000 and 5,000,000. Thus, for example, between 



