20 



REPORT 1877. 



for A .and E the bands A and A' of the foregoing description, it might be possible 

 to establish between these bands the required variable velocity-ratio. 



On the Values of a Class of Determinants *. 

 By J. W. L. Glaisher, M.A., F.B.S. 



On the Enumeration of the Primes in BurcJchardt and Base's Tables. 

 By J. *W. L. Glaisher, M.A., F.B.S. 



Burckhardt's 'Tables des diviseurs ' (1814-1817) give the least divisor of every 

 number up to 3,036,000, and Base's ' Factoren Tafeln ' (1862-65) give the least di- 

 visor of every number from 6,000,000 to 9,000,000 ; there is thus left a gap of three 

 millions for -which there are no printed tables. 



In 1871 I commenced the enumeration of the primes in the six millions over 

 which the published tables extend. The work was performed in duplicate by two 

 computers independently ; the two enumerations were then read with one another 

 and the discrepancies marked. All the doubtM numbers were then examined and 

 brought into agreement ; subsequently one of them was examined tie novo with the 

 original tables. 



A short account of the enumeration, as far as it had then proceeded, together with 

 an abstract of the results for two of the millions, was published in the British Associa- 

 tion Report for 1872 (" On the Law of Distribution of Prime Numbers," Transac- 

 tions of the Sections, pp. 19-21). I have there given tables showing the agreement 

 of the number of primes counted with the theoretical numbers derived from the 

 logarithm-integral formula of Tchebycheff and Hargreave for the second and ninth 

 millions, arranged in groups of 50,000. Soon afterwards I became acquainted with 

 the enumerations printed amongst the posthumous works of Gauss (' Werke,' t. ii. 

 pp. 436-447, 1863 j, and I found many discrepancies between these results and my 

 own. This, taken in conjunction with the great difficulty of attaining the certainty 

 of accuracy in so troublesome an enumeration, led me to lay aside the work. In 

 1876, however, I recommenced the whole again, the work being carefully performed 

 by a fresh computer, who had had no connexion with the previous enumerations. 

 The primes in the first three millions and in the seventh million were enumerated 

 de novo ; but the results for the eighth and ninth millions were only examined. No 

 errors were found in the values for the ninth million given in the 1872 paper, but 

 several were found in the second million ; they are as follows : — 



and the total number of primes in the million is 70,433 instead of 70,420. 



The number of primes in each quarter million of the six millions is shown in the 

 following Table : — 



* Th 

 niinant 



lis paper is printed, under the title "On the Factors of a special form of Deter- 

 .," in the Quarterly Journal of Mathematics, vol. xv. pp. 347-356 (1878). 



