DOMINANCE OF CHARACTERISTICS IN POULTRY. 141 
proto-, hetero-, and allogenic pairs determining it, m,, m., and m, by a 
relation of the form: 
L=am, + bm, + cm, (1) 
Using methods which are relatively much simpler than those employed 
by Professor Pearson, the value found for the coefficient of correlation 
between parent and offspring for such a character was : 
itd Ss (a — c)? _ 
~ 2(a—c)? + (a — 2b + c)? (2) 
If, now, either a = } or b = ¢, the case reduces to that of dominance, 
one of the homozygotes giving rise to the same somatic character as the 
heterozygote: this is virtually the case discussed by Pearson, and 
accordingly the value of R is the same as that found by him, viz. 4. If, 
however, the heterozygote give rise in every case to a length exactly 
intermediate between those due to the respective homozygotes, we must 
have b = (a + c)/2, whence R= 4. This is the greatest value that the 
above expression for R can attain, and consequently a character of the 
kind considered may exhibit coefficients of heredity lying anywhere 
between the limits 4 and 4, for random mating of the parents. With 
homogamy, higher values could, no doubt, be obtained. There is there- 
fore no difficulty in accounting for a coefficient of 0°5 on the theory of 
segregation, but such a value probably indicates an absence of the somatic 
phenomenon of dominance. In the case of characters like stature, span, 
&c. in man this does not seem very improbable. 
As regards the coefficients of correlation with the higher ancestry, the 
theory leads*to results which are still rather limited, for the ratio of 
successive coefficients appears to be always 4 ; 7.e. in the case of dominance 
or Pearson’s case we obtain his series 4, 7, 74, &c., and in the case of 
perfect blending the series 4, |, } &c. This second series implies, it 
should be noted, a complete absence of “ancestral inheritance’’ in the 
proper sense of the term, the partial coefficients of correlation between 
the offspring and the higher ancestry being all zero. 
A complete theory of heredity should take into account, besides 
germinal processes, the effect of the environment in modifying the soma 
obtained from any given type of germ-cell—an effect which is hardly 
likely to be negligible in the case of such a character as stature. This 
may be done without much difficulty for the limited case discussed. Let 
us suppose that the protozygote determines segments which haye not all 
the same length a, but, owing to the varying effect of environment, 
a mean length a and a standard deviation wu. Similarly, let the mean 
lengths and standard deviations of segments determined by the hetero- 
zygote and allozygote be b and v, c and w respectively. Then the value 
of R as given in equation (2) is reduced by the addition of a term 
Bu2 + 4v? + 3w? 
to the denominator. The common ratio of the ancestral coefficients 
remains, however, unaltered at its former value of 5. So far as the 
coefficients of correlation alone are concerned, it is accordingly impossible 
