1876,] of the Primes in Base's Tables. 19 



desirous of communicating the general results to the society, with- 

 out stopping for the calculation of the theoretical values, or 

 attempting a discussion. These of course will be added hereafter 

 when the whole enumeration is complete. 



Tables I, II, III which accompany this paper give the results 

 of the enumeration, the arrangement being the same as in Gauss's 

 tables for the second and third millions, except that each table 

 refers to 1,000,000 natural numbers instead of to 100,000. 



The explanation of the tables is simple. If for convenience 

 of expression we call the hundred numbers between lOOn and 

 100{n + 1), a ' century ' (so that e.g. the hundred numbers be- 

 tween 6,000,000 and 6,000,100 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 between 6,000,000 

 and 6,100,000 two centuries are composed wholly of composite 

 numbers, two centuries contain each one prime, seventeen centu- 

 ries contain each two primes, fifty-two contain each three primes, 

 and so on. Of the thousand centuries between 6,100,000 and 

 6,200,000 there is no century consisting wholly of composite num- 

 bers, there is one century only that contains one prime, &c. 



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

 of primes in the group of 100,000 to which the column has 

 reference, thus between 6,000,000 and 6,100,000 there are 6,397 

 primes; between 6,100,000 and 6,200,000 there are 6,402 

 primes, &c. 



It will be noticed that in the eighth million there are two 

 centuries that contain 15 primes; but no century in either the 

 seventh or ninth million contains so many. There are, however, 

 three centuries in the seventh million, two in the eighth, and two 

 in the ninth, each of which contains 14 primes. 



It is interesting to note how slowly the numbers of primes in 

 the successive millions diminish. The seventh million contains 

 63,799 primes, the eighth 63,158, and the ninth 62,760 : so that 

 there are only 641 primes less in the eighth than in the seventh 

 million, and only 398 primes less in the ninth than in the eighth 

 million. 



The numbers of primes in each quarter million from 6,000,000 

 to 9,000,000 are : 



seventh million. 

 First quarter 15,967 

 Second „ 15,941 



Third „ 15,950 



Fourth „ 15,941 



Total 63,799 63,158 62,760 



