18 Mr Glaisher, On the Remits of an enumeration [Dec. 4. 



of 50,000 : the second million was chosen for publication in 

 preference to the first, chiefly because results derived from the 

 counting of primes in the first million had been already pub- 

 lished by Legendre, Hargreave, and others. Soon afterwards I 

 became acquainted with the enumerations printed among the 

 posthumous works of Gauss (Werke, t. ill. pp. 436 — 447, 1863), 

 and I found many discrepancies between these results and my 

 own. This, taken in conjunction with the great difiiculty of 

 attaining not only accuracy, but the certainty of accuracy, in an 

 enumeration which is of so troublesome a kind, led me to lay 

 aside the work till 1874, when I had a considerable portion of it re- 

 computed, and satisfied myself that the enumeration found among 

 Gauss's papers contained many errors. 



Several months ago I recommenced the whole again (be- 

 ginning with the ninth million and proceeding downwards), 

 the work being very carefully performed by a fresh computer, 

 who had had no connexion with the previous enumerations. 

 The ninth and eighth millions were examined with Dase's 

 tables, and no errors were found; and from the care taken 

 throughout I have a great degree of confidence that any 

 error would have been detected : a portion of the work I did 

 myself. There was some reason to think that the original enu- 

 meration for the seventh million was not entitled to so much 

 credit as that for the eighth and ninth millions, and accordingly 

 I had this million recomputed de novo, without reference to the 

 existing enumerations. This new enumeration was performed 

 with great care, and, on comparison with the old one, it was found 

 to be absolutely free from error : there were two discrepancies, one 

 of which was due to an uncertainty in the printed table of Dase, 

 and the other to an error of a unit in the old calculation (the 

 number had originally been right, and had been altered from right 

 to wrong on the final examination). 



The enumeration for the other three millions (1 to 3,000,000) 

 is in progress ; and I propose also to calculate, probably for groups 

 of 10,000 or 50,000, the theoretical values given by the lia; 



formula, and also by Legendre's formula :; t-kf^^^t. ', takinof 



' -^ ° logic- 1-08366 ® 



account also of Riemann's investigation, " Ueber die Anzahl der 



Primzahlen unter einer gegebenen Grosse " (Bernhard Riemann's 



Gesammelte Mathematische Werke, Leipzig, 1876, pp. 136 — 144). 



The work has been so long on hand, and — however simple it 



may seem at first sight— is shown by experience to require the 



expenditure of so much care and time, in consequence of the 



extreme difficulty of attaining absolute accuracy, that, now that 



I am satisfied of the correctness of the enumeration as far as 



relates to the three millions over which Dase's tables extend, I am 



