228 
G. Udny Yule. 
for dominance , or, to define it in a somewhat more general sense than 
that used by Mendel, the condition that the hybrid zygote An should 
always behave, as regards the production of the attribute noted, as 
if it were a pure zygote of the one race, say A ; (ii.) the necessity for the 
given somatic attributes being rigidly predetermined by the characters 
of the gametes, and not liable to such fluctuations owing to the variations 
of circumstance or otherwise that an individual of the pure raced might 
be classed as an a or vice versa. As pointed out already (p. 206 ) this 
condition cannot universally hold good. To take an example from 
the inheritance of disease, the chances of an individual dying of 
phthisis depends not only on the phthisical character of his ancestry, 
but also very largely on his habits, nurture, and occupation. 
If, however, either dominance fail, or the rigid predetermination of 
the somatic attributes by the germ cell, or both, Mendel’s Laws will 
cease to hold, but the Law of Ancestral Heredity will still apply. 
Suppose first that dominance fails, and, to take a rather 
interesting case, suppose the failure to be complete, i.e. assume the 
heterozygotes on development exhibit A characters and u-characters 
with equal frequency. Then when the two forms are first crossed 
the resulting offspring, hybrid without exception, will exhibit both 
attributes with equal frequency. When the&e hybrids are crossed 
inter se, the offspring will again exhibit both characters with equal 
frequency, but one-half of both forms will be pure, the other half 
hybrids. All succeeding generations after this will be the same. 
Consider then the offspring of say 400 ^J’s. Mating being 
random, as before, conjugations of an A -gamete with an ^-gamete and 
with an a-gamete will be equally frequent. Of the 200 puree’s, 100 
mate with A' s and produce 100 pure lOOmate with a's and produce 
50 hybrid ,/s and 50 hybrid a's. The two hundred hybrid A' s may 
be treated separately as 100 pure A's and 100 pure a's. The former 
give rise to 50 pure A's, 25 hybrid A's and 25 hybrid a's, the latter to 
25 hybrid A's, 25 hybrid a’s and 50 pure a's. Adding up, the four 
hundred A's give rise on the whole to 250 A -forms (150 pure, 100 
hybrid), and 150 <2-forms (100 hybrids, 50 pure). The chance of an 
A producing an A is therefore = f. The chances of an A 
whose parent, parent and grand-parent, and so on are A's producing 
£n , 4 -form as offspring are most easily calculated by the method of 
equations (5) — (10). To use the symbols of those equations let 
T n denote the total number of J-individuals of the 7/th generation, 
all of whose ancestors in one line are also A's, and let p u of these be 
pure, i n impure. Then we have in the present case :— 
