factors. In reuniting in fertilized eggs, the chance is just half and
half that an _H_ will unite with another _H_ or with an _A_--that an
_A_ will unite with an _H_ or another _A_. Thus we have two chances of
getting _HA_ to each chance of getting either _AA_ or _HH_. Half the
zygotes will be _HA_, one-fourth _HH_ and one-fourth _AA_.

If we consider four average males, one will have two _A's_ (absence of
the factor for horns) and will thus be hornless. One will have two
_H's_, or the double factor for horns, and hence will exhibit horns--as
will also the two _HA's_ since a single dose of horns expresses them in
a male. So we have the three-to-one Mendelian ratio.

But four females with exactly the same factors will express them as
follows: The one _HH_ (double factor for horns) proves sufficient to
express horns, even in a female. The _AA_, lacking the factor entirely,
cannot have horns. Nor will the two _HA_ females have horns, a single
dose being insufficient to express them in a female. Again we get our
three-to-one Mendelian ratio, but this time it is three hornless to one
horned.

Especially Goldschmidt's carefully graded experiments point to a similar
difference in the strength of the dose or doses of the sex factors.
Instead of the two doses of horns required to express them in the
presence of the female secretory balance in Professor Wood's sheep,
Goldschmidt found it took six doses of maleness to completely express it
on a female basis in his moths. But even with three doses, the female
was incapable of reproduction. A single dose in excess of the ordinary
combination to produce normal females modified the type of body, also
reducing the number of eggs.