CHAP, xiii] FACTOB TABLES, LISTS OF PRIMES. 353 J. W. L. Glaisher67 gave for the second and ninth millions the number of primes in each interval of 50000 and a comparison with lix' — lix, where lix=fax/log x [more precise definition at the end of Ch. XVIII]. A committee68 consisting of Cayley, Stokes, Thompson, Smith, and Glaisheri prepared the Report on Mathematical Tables, which includes (pp. 34-9) a list of factor and prime tables. J. W. L. Glaisher69 described in detail the method used by his father70 and gave an account of the history of factor tables. Glaisher690 enumerated the primes in the tables of Burckhardt and Dase. Glaisher696 tabulated long sets of consecutive composite numbers. He69c enumerated the prime pairs (as 11, 13) in each successive thousand to 3 million and in the seventh, eighth, and ninth millions. E. Lucas69rf wrote P(q) for the product of all the primes g #, where q is the largest prime < n. If zP(#)=*=! are both composite, xP(q)—n,..., xP(q),. .., xP(q)-{-n give 2n+l composite numbers. Glaisher696 enumerated the primes 4n-f-l and the primes 4n+3 for intervals of 10000 in the Hh million for fc = l, 2, 3, 7, 8, 9. James Glaisher70 rilled the gap between the tables by Burckhardt45 and Dase61. The introduction to the table for the fourth million gives a history of factor tables and their construction. Lehmer92 praised the accuracy of Glaisher's table, finding in the sixth million a single error besides two misprints. Tuxen71 gave a process to construct tables of primes. Groscurth and Gudila-Godlewksi, Moscow, 1881, gave factor tables. *V. Bouniakowsky71a gave an extension of the sieve of Eratosthenes. W. W. Johnson716 repeated Glaisher's70 remarks on the history of tables. P. Seelhoff72 gave large primes k-2n-\-l (/c<100) and composite cases. Simony73 gave the digits to base 2 of primes to 214 = 16384. L. Saint-Loup74 gave a graphical exposition of Eratosthenes' sieve. H. Vollprecht76 discussed the construction of factor tables. ''Report British Association for 1872, 1873, trans., 19-21. Cf. W. W. Johnson, Des Moines Analyst, 2, 1875, 9-11. "Report British Association for 1873, 1874, pp. 1-175. Continued in 1875, 305-336; French transl., Sphinx-Oedipe, 8, 1913, 50-60, 72-79; 9, 1914, 8-14. MProc. Cambridge Phil. Soc., 3, 1878, 99-138, 228-9. MiJbid., 17-23, 47-56; Report British Assoc., 1877, 20 (sect.). Extracts by W. W. Johnson, Des Moines Analyst, 5, 1878, 7. •^Messenger Math., 7, 1877-8, 102-6, 171-6; French transl., Sphinx-Oedipe, 7, 1912, 161-8. "e/Wd,, 8, 1879, 28-33. Md/Wd., p. 81. C. Gill, Ladies' Diary, 1825, 36-7, had noted that xP(q)+j is composite for j=2,...,<z-l. B9eReport British Assoc., 1878, 470-1; Proc. Roy. Soc. London, 29, 1879, 192-7. 70Factor tables for the fourth, fifth and sixth millions, London, 1879, 1880, 1883. 71Tidsskrift for Mat., (4), 5, 1881, 16-25. naMemoirs Imperial Acad. Science, St. Petersburg, 41, 1882, Suppl., No. 3, 32 pp. 7lbAnnals of Math., 1, 1884-5, 15-23. 72Zeitschrift Math. Phys., 31, 1886, 380. Reprinted, Sphinx-Oedipe, 4, 1909, 95-6. "Sitzungsber. Ak. Wiss. Wien (Math.), 96, II, 1887, 191-286. 74Comptes Rendus Paris, 107, 1888, 24; Ann. de l'6cole norm., (3), 7, 1890, 89-100. "Ueber die Herstellung von Faktorentafeln, Dies. Leipzig, 1891.