On Mendel's Laws. 
229 
Pn + 1 = £ Pn + i in ( 1 0 
in + 1 — 4 Pn + 4 ^ — T T„ ( I2 ) 
whence, as before, if C„ be the chance of an ^-individual, all of whose 
n ancestors in one line are A’s, producing an /1-form as offspring 
C _ 3. 
n - 4 
1 
16. C 
03 ) 
n x 1 
This equation gives the following figures for the successive 
chances:— 
Cj = *62500 
C, = *65000 
C 3 = *65385 
C 4 = *65441 
C 5 = *65449 
where again the value of C tends towards a fixed limit *6545085 ... 
obtained by writing C n for C n _, in equation ( 13 ) and solving. As we 
have assumed the complete absence of dominance in the above case, 
the values of C given will apply to both a ’s and A’s. - Had we 
assumed only a partial failure of dominance, supposing, e.g. the 
heterozygote to give rise to 80$ of H’s and 20$ of < 7 ’s, the law of 
ancestral heredity would still have applied to both forms, but the 
two series of chances would have been different. The failure of 
dominance thus generalises the forms of the law of ancestral 
heredity derivable from Mendel’s principles in two different 
ways (i.) by rendering it possible to obtain any arbitrary value for Cp 
(ii.) by rendering the law applicable to a’s as well as A’s. 
I have, of course, in the preceding, assumed random mating 
merely to simplify the work. If the mating were homogamic, A’s 
only mating with A’s, and n’s with n’s, as would generally be the 
case if the breeder wished to obtain as nearly as possible pure races 
of A’s and < 7 ’s, the law of ancestral heredity would still apply. The 
working would, however, be a good deal complicated, for the pro¬ 
portion of pure forms and hybrids amongst the A’s would vary from 
one generation to the next. 
Next suppose the absolute predetermination of the somatic 
attributes by the germ-cell to fail, pure zygotes of the one type 
producing not only A’s, but also a proportion of a’s, and pure zygotes 
of the other type, not only a’s, but also a proportion of A’s. At the 
same time, to keep the case fairly simple, let dominance in a 
generalised form still hold good, the heterozygote behaving, as 
regards the proportions of the two forms produced, precisely as if 
it were a pure homozygote of one or other type. Let the two classes 
of gametes be, say, B's and b’s, pure ZLzygotes producing 70 % of 
