H. L. RiETZ 
117 
With the formula for the selection of one character only, we obtain 0"695 as 
the corrected correlation of age and production from 0"662. 
It should be understood that these corrected values are to be regarded as 
approximate results only. In finding the percentage of cows that meet the 
standards, I think perhaps our method gives values for these percentages rather 
too small than too large, owing to the fact that breeders are likely to keep from 
the test animals that can barely meet the requirements. If this be the case, our 
corrected values of standard deviation and correlation are somewhat larger than 
they should be. 
(9) Mature and immature cows. To return to the consideration of mature 
and immature cows (§ 4), it appears that, both before and after applying the correc- 
tion for selection, the coefficient of heredity is greater for mature than for very 
young cows. We shall next make use of the partial correlation coefficient to 
obtain a sort of average value for the correlation coefficient during the period of 
growth. Following the usual notation, let the subscripts 1, 2, 3, 4 mark produc- 
tion of offspring, age of offspring at time of test, production of dams, and age of 
dams at time of test respectively. Then the desired value is given by 
p ^ (1 - r^i) + ^2 {rar^i - r^) + {r^^r.^ - r^) 
Jl- r.J - roj - r^^ + 27\,ruro^ Vl - r,,- - r,.^ - r^' + 2r,.r,,r^ ' 
The values to be substituted in this formula are ?'i3 = 0'18 from the material of 
Table IX., corrected for selection ; r^o — r^i - 0'G95 from Table I. corrected for 
selection; = 0-033 from Table XIII.; ru = 0, n3 = 0. 
Pi3 = 0-38. 
We have then O'SS for a sort of average value of the correlation coefficient at 
fixed ages in the period of growth, 0"145 and 0'25 for uncorrected and corrected 
coefficients of cows under 2-25 years old, 0'28 and 0"63 for uncorrected and corrected 
coefficients after maturity. These results seem convincing of the fact that inherit- 
ance in production of butter is, in general, much more pronovnced> after maturity 
than at a fixed stage in the period of grorvth. 
(10) Summary of results. 
(a) The correlation of age and production of butter fat by cows under 4-75 
years is approximately f , and the regression is linear. 
(h) The correlation of age and production of cows over 475 years is sub- 
stantially zero. 
(c) The coefficient of variability in production increases gradually from youth 
to three years old, and then decreases to maturity. 
(d) Production of butter fat is a function transmitted to a marked extent. 
The coefficient of heredity between cows and dams in the performance of this 
