225 



On the Ancestral Gametic Correlations of a Mendelian Population 



mating at Random. 

 By Karl Pearson, B.R.S. 

 (Eeceived April 2,— Read April 22, 1909.) 



(1) The population to be considered in this paper is supposed to be 

 initiated by a group of S\ individuals with the protogenic constitution (AA), 

 s 2 individuals with the allogenic constitution (ad), and s 3 individuals with 

 the hybrid constitution (A<z), where the mating is given by the simple 

 Mendelian formula: (AA) x (ct«) = 4(Aa). I do not assume at this stage 

 any relation between the gametic constitution of an individual and its- 

 somatic character. I propose first to consider the correlation between any 

 ancestor and the resulting array of offspring, when we regard only their 

 gametic constitutions. I assume that all mating in the population is 

 random, i.e. that every possible mating occurs simply in the proportions of 

 the frequency of individuals of given gametic constitution in the population, 

 and that there is no differential fertility or selective death-rate. 



In a paper published in the ' Phil. Trans.,' vol. 203, A, 1904, p. 53 et seq.,. 

 I have dealt with the correlation between the somatic characters of the 

 ancestry and the offspring in a population of a Mendelian character, more 

 general in that I supposed the character to depend upon n couplets, and not 

 a single Mendelian couplet, less general in that I supposed the population 

 to have arisen from a series of initial hybridisations, and not from a 

 mixture as in the present case of hybrids and members of two pure races- 

 in any proportions. In that paper I showed (a) that there was correlation 

 between any ancestor and the offspring, (b) that the regression for any 

 ancestor and the offspring was linear, and (c) that the correlations decreased 

 in geometrical progression. These are the chief characteristics of the Law of 

 Ancestral Heredity. It was clear that, judged by somatic characters only, 

 ancestry was of importance. The result depended on Mendel's first principle 

 of dominance being absolutely true. The values of the correlations were, how- 

 ever, less than those with which biometric work had made us familiar. 



(2) In the present paper I start with a more general population, and 

 investigate the correlation of the gametic not the somatic characters. 



The general formula for the population before the first mating is 



S!i(AA) + 2s 3 (Aa) + s 2 (aa). (i> 

 After the first random mating it is 



( Sl + ,s 3 ) 2 (AA) + 2 (. Sl + s 3 ) (s 2 + s 3 ) (Aa) + (s 2 + s 3 f (aa). 



