448 On the Itiheritance of Coat-Colour in Cattle 
beasts with no whole red, the latter class including roans and whites. Table IV. 
gives our results : 
TABLE IV. 
Parental Inheritance in Shorthorns. 
Nature of Percentage 
Mean Square 
Contingency 
Mean 
Contingency 
Fourfold 
Table 
Sire and Bull Calf ... 
■37 
•43 
•46 
Sire and Cow Calf ... 
■40 
•50 
•52 
Dam and Bull Calf ... 
■40 
•44 
•44 
Darn and Cow Calf ... 
•42 
•46 
•43 
Mean 
•40 
•46 
•46 
The result seems to us to indicate that the values by mean square contingency 
have come out somewhat too small. The general effect of an assortative mating 
if there were no selection of parents would be to increase the apparent parental 
correlation. If r were the real correlation, p the assortative mating, and X the 
observed parental correlation, we should have 
1 + p 
X = r~. ^ =^. 
Vl + 2/3r- 
If r = •S and p = "2, we should expect the correlation altered by about -^^ 
increment. But this assumes that there has been no selection of parents, 
whereas in the case of the bulls at least there is obviously very considerable 
selection. The tables show that the red whites of both kinds are rejected, and 
that red and roan bulls are the most fashionable. But we know that any 
stringency in parental selection is more effective in reducing parental correlation 
than assortative mating can be in raising it*. The amount of this cannot be 
determined in the present case, but it very probably contributes to the reduction 
observed in the above table below the values determined for horse and grey- 
hound. 
The general conclusion that we can draw is that the parental resemblance 
to offspring in shorthorn cattle exceeds ^4, and may quite reasonably equal "46, 
or even a higher value. It lies therefore well within the range of values "4 to "5 
found already for man, horse, and dog. — 
(9) On Direct Inheritance. Second and Higher Degrees. 
We now turn to the degree of resemblance between the offspring and more 
distant ancestry. The following results were obtained "I*: 
"Mathematical Contributions to the Theory of Evolution," Phil. Trans. Vol. 200 A, p. 39. 
t The tabulated material will be found in the Appendix as Tables H, I, J, K, L, M, N, and 0. No 
individual grandparent was taken more than six times, and on the average no individual occurs more 
than three times. 
