194 



Mr. J. W. L. Glaisher. 



[June 19, 



counted as a 4^ + 3 prime, or vice versa; but pains were taken to 

 verify completely that the pencil lines had been ruled correctly, and 

 for two of the millions (the seventh and ninth) the ruling was per- 

 formed twice independently, in the one case a second copy of Dase 

 being used, and in the other the original lines having been rubbed 

 out before the second calculation was commenced. The enumerations 

 were however troublesome, and it was much more difficult to ensure 

 freedom from error than in the case of the simple enumerations of 

 primes. 



A specimen of a portion of one of the printed forms for the entry of 

 the number of primes in each hundred, which have been referred to 

 above, is given on p. 102 of vol. vii of the "Messenger of Mathe- 

 matics "* (1877), and a preliminary account of the results of the 

 complete enumeration of all the primes in the first, second, third, 

 seventh, eighth, and ninth millions has been printed in the " Pro- 

 ceedings of the Cambridge Philosophical Society," vol. iii, pp. 17-23, 

 47-56 (1876-77). f The results of the enumeration for the fourth 

 million will appear in the volume containing the factor table for that 

 million. 



The total numbers of primes of the forms 4?z, + l and 4?i + 3 in the 

 first one hundred thousand numbers of each million, taken from the 

 subjoined tables, are 







Number of 



Number of 





Total 







4n + 1 



4» + 3 



Difference. 



number of 







primes. 



primes. 





primes. 



0- 



- 100,000 



4,781 



4,808 



-24 



9,592 



1,000,000- 



-1,100,000 



3,642 



3,574 



+ 68 



7,216 



2,000,000- 



-2,100,000 



3,463 



3,411 



+ 52 



6,874 



3,000,000- 



-3,100,000 



3,368 



3,308 



+ 60 



6,676 



6,000,000- 



-6,100,000 



3,193 



3,204 



-11 



6,397 



7,000,000- 



-7,100,000 



3,182 



3,187 



- 5 



6,369 



8,000,000- 



-8,100,000 



3,126 



3,124 



+ 2 



6,250 



The most noticeable feature in these numbers is the excess of the 

 4re + l primes over the 4/^ + 3 primes in the groups taken from the 

 second, third, and fourth millions ; in these three groups there are 

 10,473 primes of the form 4/z + l, and 10,293 of the form 4??, + 3, the 

 difference being 180, and the total number of primes 20,766. 



* "On Long Successions of Composite Numbers," vol. vii, pp. 102-106, 171-176. 



f An enumeration of prime-pairs (i.e., of pairs of primes separated by only one 

 number), is given in the " Messenger of Mathematics," vol. viii, pp. 28-33 (1878) ; 

 and a full account of Burckbardt's and Dase's tables occurs in a paper " On Factor 

 Tables, with an account of the mode of formation of the Factor Table for tbe Fourth 

 Million" (" Proceedings of the Cambridge Philosophical Society," vol. iii. pp. 99- 

 138, 228-229, 1878). See also " British Association Keport," 1878, pp. 172-178. 



