By John Burdon Sanderson Haldane

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Crozier and Snyder. 1922. Proc. Soc. Exp. Biol. Med. 19, 327. Detlefsen. 1915. Genetics 3, 573. Durham. 19il. J. Genetics 1, 107. Goldschmidt. 1920. Z. indukt. -u. Verblehre 23, 1. Hardy. 1908. Science 28, 49. Heribert-Nilsson. 1920. Hereditas 1, 41. Ibsen and Steigleder. 1917. Am. Nat. 51,740. Jennings. 1915. Genetics 1, 53. Jones. 1920. Biol. Bull. 311,251. Morgan and Bridges. 1916. Carn. Inst. Wash. , 237. Pearl. 1907. Biometrika 5, 213. Pearl. 1912. Am Nat. 46, 130. Pearson. 1904. Phil. Trans.

Selection therefore proceeds much as according to equation 72 of Part I. (b) Stable equilibrium, 0 > 2K+ k', 2k + k' > 2K. 2K-2k-k' • • 3 niOS ^ {) n /ti»\ 2k-4>K} + l (2Eun+k'un-2K+ 2k+k'\ O0 0 g (2-2) ) where u0 5 ux according as un 5 vm. The results of Robertson (3) suggest that milk-yield in cattle depends on one or more sex-linked factors which act most effectively when heterozygous, besides autosomal factors. If so human effort in this case has given K a negative value, while k and k' are nearly zero.

A. FISHER. Proc. Roy. Soc. , vol. 42, p. 321, 1922. E. ROBERTSON. Journ. Genetics, vol. 11, p. 79, 1921. H. NILSSON-EHLE. Lund's Univ. , p. 69, 1909. E. R. SADNDERS. Journ. Genetics, vol. 10, p. 149, 1920. 6. GREGORY. Proc. Roy. Soc, B, vol. 87, p. 484, 1914. 7. BLAKESLEE, BELLING and FARNHAM. Science, vol. 52, p. 388, 1920'. Mr Haldane, A mathematical theory, etc. 607 A mathematical theory of Natural and Artificial Selection. Part IV. By Mr J. B. S. HALDANE, Trinity College. ith, and the resulting finite difference equation investigated.