492 
Proceedings of the Royal Society of Edinburgh. [Sess. 
giving the same correlation coefficient as before, namely, 
T (8 + 4m - m 2 )* 
With the increase of m the correlation becomes less. The values of r 
are given under Case (a). 
Correlation Coefficients when more than two Races Mix. 
33. So far, a mixture of two races alone has been considered. Many 
stocks of cattle, etc., are supposed to be derived from more than two, so 
that a brief consideration of how this affects the correlation values is 
necessary. With the same notation let the original races be — 
(a, a) (b, b) (c, c). 
Then the stable population with random mating is as before, 
(a, a) + (b, b) + (c, c) + 2 (a, b) + 2 (a, c) + 2 (b, c). 
A correlation table is then easily written down and is as follows : — 
Parent. 
Offspring. 
(a, a). (a, b). 
(b, b). (b 3 c). 
(c, c). (c, a). 
(a, a) . 
3 3 
3 
(a, b) . 
3 6 
3 3 
3 
(b, b) . 
3 
3 3 
(b, c) . 
3 
3 6 
3 3 
(c, c) . 
3 
3 3 
(c, a) . 
3 3 
3 
3 6 
To evaluate the correlation the product method hitherto used is 
inapplicable, and the method of contingency must be employed. In the 
first place, on the supposition that all hybrids are distinct, we have r = , 59 l 7, 
which is considerably higher than the value r='487, found by contingency 
when only two types of parent are considered. 
Secondly — 
34. If a be dominant over b, b over c, and c over a (indicated in table by 
dotted lines), the coefficient when estimated by mean square contingency 
falls in value to '425. This case is very suitable for a fourfold division, 
