112 
Heredity in Production of Butter Fat 
These results agree satisfactorily with those drawn from Table III., where the 
relationship is the same as in this case. 
With transmuted values, we obtain the following coefficients of correlation 
(Tables VI., VII.): 
Maternal Granddams and Offspring, r = 0 138 ± 0-024. 
Paternal Granddams and Offspring, r 0-086 + 0 017. 
While it is impossible to measure the potential character of butter fat pro- 
duction in the male, we aim to determine the extent to which he transmits this 
character from his dam to his female offspring. It may be observed that the 
coi relation coefficient of maternal granddams and offspring is 0-138 while that of 
paternal granddams and offspring is 0-086. It is, however, to be further observed 
in this connection that the table of paternal granddams contains a considerable 
number of scattered extreme variates. If these were omitted the correlation 
coefficient would be appreciably increased. Furthermore, aside from entrance 
requirements, there is a strong tendency to select paternal granddams of high 
production so that selection may enter as a factor. For these reasons, we hesitate 
to attach significance to the difference 0-138 — 0-086 = 0-052 although possibly it 
indicates prepotency of females over males in the transmission of this character. 
(6) Selection of Sires. 
In breeding problems, the artificial selection of sires is comparable to assortative 
mating in man. The question thus arises : To what extent is the production of 
butter fat by any cows correlated with the production by dams of sires to whom 
the cows are mated ? In answer to this question, we obtain the result (Table 
VIII.) 
r = 0-140 + 0-025 
for the correlation in question. 
(7) Further Correlations for immature Cows. 
In beginning this work, I expected to apply in a straightforward way the 
theory of multiple correlation by carrying the work back to granddams, and at 
the same time to take the age element into account. Then, in the usual notation, 
the regression equation could be written in the form 
= a-i,20c^ + a-^^x^ + a^iX^ + a^^x^ + a^x^ + al^x^ + a^^x^, 
where x^, x^, x^, Xj refer to production of offspring, dams, maternal granddams, 
and paternal granddams respectively; and x\, x^, Xg, Xs refer to ages at times of 
respective tests of offspring, dams, maternal granddams, and paternal granddams. 
In Tables IX. to XII. is collected material for the correlation of first records of 
offspring with most remarkable records of ancestors when no regard is given to 
age except that all animals are under five years old. From Table IX., we obtain 
for the correlation of offspring and dams 
= 0-132 ±0-023. 
