1903.] Generalised Theory of Alternative Inheritance, 505 



"Mathematical Contributions to the Theory of Evolution. XII. 

 -On a Generalised Theory of Alternative Inheritance, with 

 Special Eeference to Mendel's Laws." By KARL PEARSON, 

 F.E.S. Received September 1 1, Read November 26, 1903. ' 



(Abstract.) 



(1). A great deal of interest has been recently excited by Mendelian 

 theories of inheritance, and a considerable amount of controversy has 

 arisen with regard to their relation to the statistical treatment of 

 inheritance within large populations. The fundamental conception of 

 the Mendelian theory is the non-fusal of the gametes of two pure 

 races on crossing. If A, A' represent the pure gametes of one race and 

 a, a those of the other, the individual arising from the zygote formed 

 by crossing these races contains not fused gametes, but pure gametes 

 of the two original races, and these gametes are equally represented 

 on further crossings among the hybrids. The whole may be exhibited 

 symbolically as 



(AA') x (ad) = (A + A') (a + a') = (Aa) + (A.a r ) -f (A'a) + (aa') t 



where in the first product we have the parental zygqtes ; in the .second 

 product the breaking-up into the gametes is indicated, and on the 

 extreme right the zygotes of the resulting offspring are given. This is 

 a perfectly simple conception. We can generalise it at once by 

 replacing the simple couplet (AA') by n such .couplets, and supposing 

 the original parental zygotes to be of the form 



(AiAi) + (A 2 A 2 ) + (A S A 3 ) + . +(A n A n ), 



where each corresponding pair of couplets on crossing obeys the above 

 symbolic formula. If two individuals, constituted as above, be crossed, 

 they give a generation of hybrids. If these hybrids now breed at 

 random and are equally fertile among themselves, segregation takes 

 place. If the process of random mating with equal fertility be 

 continued generation by generation, what further changes, if any, take 

 place, and what are the laws of inheritance within such a population 1 

 This is the general problem I have set myself to attack by mathematical 

 analysis. I speak of it as a generalised theory of alternative inheritance ; 

 it is based on a pure gamete conception. It is not Mendelian in so far 

 as it replaces his simple couplet by ^-couplets, and supposes the 

 character determined by these ^couplets and not a sing e couplet 

 Further, it entirely dispenses with Mendel's fundamental ideas of 

 recessive and dominant characters. To avoid any such assumption I 

 speak of the protogenic and allogenic couplets (aa) and (AA).^ I 



VOL. LXXJL 



