4 
178 Miscellanea 
Hence if we assume the mean of array of offspring to he given by 
1 /.?; 
2 \(Ti ^ (72 
(i) the second portion of the expression must be zero, i.e. mean of whole population of 
offspring must coincide with mean of array of ofl'spring where parents have the mean values and 
(ii) we must have ''i:! = >'23 = i- Iii other words the form of our assumption involves both the 
equal influence of the parents and the value of the parental correlation. 
From the standpoint of hereditj^ no such a.ssumption is legitimate. Neither in Mendelian 
theory nor in biometric formula, nor again in actual observation is it permissible to suppose that 
the mean of the array of offspring is determined solely by the parents. Still less is it possible 
to sup2)ose the actual character of the oflfspring to be the mean of that of the parents (i.e. put 
«=0). If it were we shoiild liavo z = h (*'+»/), whence flow 
.(xiv). 
,„ 1 , , 
=-^{i^2+ M2 ) 
M,s"' = ^ (Ms' + Ms") 
^4'" = ^ if^i + 6fi2V2" + M4") ^ 
But these equations assume that /xj'^, fis" and ^4'" are all zero — an absurdity in itself and 
contrary to all experience, whether biometi'ic or Mendelian. For non-as.sortative mating and 
equal potency of parents, they lead to parental correlations of the order and to an impossibility 
of stability in any pojjulation*. 
In fact any such relations as (xiv) are inconceivable on the basis of both biometric as well as 
Mendelian theory and observation. Parental correlations have never been observed anywhere 
near such a value as 0 7. Equations (xiii) are, however, suggestive ; they show that if the 
jiarental distribution be sj'mmetrical and mesokurtic, the array of ofispring will i-emain so after 
selection ; but if the parental distribution does not possess these characters, then any selection of 
individual parents will emphasize the asymmetry and the kurtosis in the resulting array of 
ofispring ; or continued selection of this type will lead to greater and greater divergence from the 
normal or Gaussian frequency distribution. 
* If we assume that the mean of the array of offspring of parents of characters x and y is given by 
Ix + my, it is only another way of asserting that the regression is linear and that 
I _ Oj _ ?j3 - »-i2 '•23 03 
1 - ci ' 1 - rjs^ (T'> ' 
If we make l = m, or give equal weight to the parents, it is on]y rational to suppose that a-i = a2 and 
ri2 = , which lead us to 
1 + r-,3 (T, ■ 
Hence the mean of the array is — ?J ix-i- 11] 
1 + 7-23 <Xi 
and whether we make x constant and y constant 01 x + y constant leads to precisely the same variability 
in the array, i.e. 
, ^ /l-n2^-n3^-^^ + 2ri2ri3r23 / 27^12^" 
V l-r23^ V l+»-23 
If assortative mating be zero, this equals 
0-3 »y 1 - 2ri2^ 
and, if to reach the results for ^i'" given above we put this zero, we must have 
ri2=\/-50 = 0-7 nearly. 
