452 
Oyi the Inheritance of Coat-Colour in Cattle 
But we do not consider these results as they stand as strictly comparable. We 
believe the lower values in the case of the shorthorns are not due to a lesser 
intensity of inheritance, but to the fact that the only possible contingency table is 
not finely enough divided. This we hold is indicated by the fourfold and mean 
contingency methods giving sensibly higher values than the mean square con- 
tingency method (see p. 448). 
It appeared desirable to determine the best geometrical series : a, ar, ar^, ar^, 
to fit the shorthorn results. Accordingly we made : 
(•40 - af + (-20 - arY + (-17 - arj + (12 - ar^f 
a minimum. This led to the least square type equations : 
•40 + •20r + ■llr-' + •12r3 = a (1 + + + r«), 
•20 + •34r + •36?-- = a (r + 2r3 + 3r^). 
Eliminating a, r is given by : 
•I7r^ + •28r« + 1-20? -' - •04r* + •63r= - •36r= + •06r - "20 = 0. 
The appropriate root of this is r = '64 nearly, giving a = •382, and the best 
fitting geometrical series is '38, "24, •IG, "10, as against the observed "40, '20, 
■17, ^12. But the remarkable result of the investigation is that the factor of 
reduction for the geometrical series in the case of eye-colour in man was found* 
to be "62, and for coat-colour in horses •66'f , and the reducing factor for shorthorns 
is '64, exactly their mean. The reducing factor for greyhounds is seen to be about 
•62|. Thus disregarding absolute values we see that the manner in which ancestral 
resemblance decreases with ascent is very closely the same for such different 
species as man, horse, greyhound, and shorthorn, and for most purposes may be 
taken equal to the round number 2/3. If we assume r = 2/3, we find the best 
value of a for the shorthorn is almost exactly '37, giving the series •37, "25, "16, 
and against the observed •40, ^20, •17, and ^12. It would thus appear that the 
general effect of applying the 25-fold mean contingency method has not been to 
lower the relative intensity of the successive grades of ancestry, but to weaken in 
about the same proportions the absolute values. The factor of reduction is about 
38/50, or we should have for shorthorns the best fitting series, 
•50, -32, 21, -13, 
as against an observed series, 
•53, •26, -22, -16. 
Either or both of these are in quite reasonable agreement with what has been 
observed for man and horse. 
Generally we may conclude that ancestral influence in the shorthorn diminishes 
at the same rate sensibly as in man, horse, and dog, and that absolute values are 
* Biometrika, Vol. ii. p. 222. 
t Ibid. Vol. II. p. 222. 
X The mean of Dr Warren's rather irregular results for Hyalopterus and Daphnia is again 'Gl. 
Biometrika, Vol. i. p. 139. 
