

APPENDIX 707 



When expanded and expressed in terms of r's and <r's, (2) becomes 



The formulas used in the text in discussing biparental inheritance are 

 special cases of (i) and (3) just derived. This may be verified by making 

 the following substitutions : 



Put x = h v y = h^ 2 = h z , r yz = r 2 , r xx = r v r xy - r s , a-* = <r 3 , v x = <r v 

 Then in the new notation (i) becomes 



Since in the case discussed in the text the parents were taken equi- 

 potent, r x = r 2 , and by making this substitution in (4) we get 



which is the formula used in text. Likewise, if we make these substitutions 

 in (3), we get for the variability of an array of sons 



2.r 



which is the formula used in the text. 



More than three variables. It is easily seen that the methods employed 

 in the case of two and three variables can be extended to any number of 

 variables. However, the complexity of the algebraic expressions becomes so 

 great that it does not seem well to present a more extended discussion here. 

 For the general case of any number of variables, the reader with consider- 

 able mathematical training is referred to the treatment by Karl Pearson in 

 the Philosophical Transactions of the Royal Society, A, CLXXXVII, 1896, 

 and A, CC, 1903. In the papers just referred to the general expression is 

 also given for the variability of an array in the case of any number of 

 variables. It is from this general expression that the formula used in the 

 text for the variability of an array of offspring after n generations of 

 selection is derived. 



Formula for the correlation coefficient r which is better adapted to numer- 

 ieal calculation. In the first place, the calculation of the standard devia- 

 tions of both systems of variates should be done by the shorter method 

 presented on page 429. 



