226 
G. Udny Yule . 
of their producing a dominant form would he 
while the chance of a dominant of unknown parentage producing 
a dominant form is only But this is precisely a case of the law 
of ancestral heredity ! It is not difficult to continue the calculations 
on the same simple lines, but the work may be abbreviated by 
the following considerations. Let T n denote the total number 
of dominants in the ;*th generation all of whose ancestors in 
one line are also dominants, and let p„ of the T n be pure, i n impure 
or hybrids. Then quite generally one-half of the pure dominants 
and one-quarter of the hybrids of any generation give rise to 
pure dominants as offspring, while the remaining half of the pure 
dominants and one-half of the impure give rise to hybrid forms. 
That is in symbols 
Pn + l = l Pn + i in (5) 
in + l = i Pn + I in = J T n (6) 
Adding the two equations together 
i ; 
4 *n 
T n + i = T„ 
or by equation (6) 
T — T _IT 
1 n + 1 1 r. 8 1 n - 1 
Dividing out on both sides of this equation by T n and writing 
( 7 ) 
( 8 ) 
n + 1 
where C n is the chance of a dominant form of the ?/th generation 
producing dominant offspring, we have finally 
q — 1 i 
U “ 1 8C 
n -1 
(io) 
an equation which enables us 
to calculate the remaining chances 
very easily, given that C l = f. 
I find ' 
C, 
== -83333 
C 2 
= -85000 
C 3 
= -85294 
C 4 
= -85345 
C 5 
= -85354 
where, as is found by equating C n to C n _! in equation (io) C tends 
towards the limiting value ’85355339 . . . The figures illustrate as 
nicely as could be desired the two chief properties of Ancestral 
Heredity—(i.) the chance of an A producing an A is increased if the 
ancestry be also A’s. (ii.) it is not of much use to take into account 
more than the first few generations of ancestry (cf. p. 203 supra), 
for the chance C rapidly approaches a limiting value. 
Mendel’s Laws, so far from being in any way inconsistent with the 
