78 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



or 



m 4-H = V -(f> T>0. 



1 + v v 



(11.) PROPOSITION IX. To find the General Formula for Biparental Regression on 

 the Theory of the Pure Gamete, and the Value to be c/iren to the " Midparent." 



If we applied without further consideration the general formula for biparental 

 regression to this case, we should have, if m-f,,/ be the mean of the offspring due to 

 fathers of /^-allogenic couplets, mated with mothers of '/-allogenic couplets, 



JL I i _ / \ . \ i i / i \ 



P'l 4 I" 3V / 4" / "I 3 W 4 /' 



This follows at once, since the mean of the general population = %n, the regression 

 coefficient for either parent = ^, and there is no assortative mating. 

 Flence we should have 



Now suppose both parents of pure allogenic race, then /> = <y = //, and all the 

 offspring will be of pure allogenic race, or we must have m^ n. 

 But the above formula gives 



/, == i, 



which is not correct. 



In other words, while the above formula gives the best plane to fit the array of 

 points determined by the parental constitutions, that array of points does not truly 

 lie in a plane. Or, although the simple regressions are linear, the compound 

 regression is not truly planar. We have therefore to find its true form, and measure 

 the amount of deviation from the truth involved in using a biparental formula of the 

 type indicated. 



Given a character resulting from n couplets, we require 4" X 4" individuals, 4" male 

 and 4" female, to form the whole possible system of random matings. In such a 

 population there would be 



<'//. /:,<y~'' fathers of /^-allogenic couplets, 

 and 



</ ../. :5 "~' / mothers of ry-allogenic couplets. 

 The chance therefore of a mating of a ^-allogenic father and a cy-allogenic mother is 



''//,/)," '','/, 3~"~ y '~' / /4 :J " ', 



and when n be even moderately large, this gets very small if p and q at all nearly 

 approach n. For example, if n were 5, and father and mother were both pure 

 allogenes, the chance of such a pure allogenic mating would be only 



1/1,048,576, 

 or m a population of a million would hardly occur once. 



