ON MATHEMATICAL TABLES. 



47 



have been made, and the factors obtained by the multiple method are in 

 course of being entered. 



The mode of construction was described in last year's repoi't (Dublin, 

 1878, pp. 172-178), and a more complete account will appear in the 

 introduction to the factor table for the fourth million. 



Results of an Enumeration of the Primes in the Fourth Million. 



Two independent enumerations of the primes in the fourth million 

 were made, one from the manuscript before it was sent to the printer, and 

 the other by a different computer from the proof sheets. The results are 

 shown in the following table : — 



The explanation of this table is as follows : — Calling, for convenience 

 of expression, the hundred numbers between lOOn -1 and 100 (n + 1) a 

 century (so that, e.g., the hundred numbers between 2,999,999 and 

 3,000,100 form a century), then the table shows the number of centuries 

 in each group of 100,000 which contain no prime, the number of 

 centuries each of which contains one prime, the number of centuries 

 each of which contains two primes, &c. Thus of the thousand centuries 

 3,000,000-3,100,000 no century is composed wholly of composite num- 

 bers, two centuries contain each one prime, eleven centuries contain each 

 two primes, thirty- seven centuries contain each three primes, and so 

 on. Of the thousand centuries 3,100,000-3,200,000, one consists wholly 

 of composite numbers, three contain each one prime, &c. 



The numbers at the foot of each column give the total number of 

 primes in the group of numbers to which tbe column has reference ; 

 thus between 3,000,000 and 3,100,000 there are 6676 primes; between 

 3,100,000 and 3,200,000 there are 6717 primes, &c. Similar tables to the 

 above for the first, second, third, seventh, eighth, and ninth millions have 



