230 
G. Udny Yule. 
A’s, pure ^-zygotes 35^ of A' s, and let the 12-gamete he dominant 
over the i-gamete in the sense that the Bb zygote produces 70% of 
A’s just as if it were a pure BB- zygote. Now, the assumption of 
the break-down of “ predetermination ” brings in a new element; 
we have not only to consider A’s developed from the dominant 
BB -zygotes and hybrid A’s developed from the heterozygotes 
but also A’s that have developed from the pure recessive bb-zygotes > 
“aberrant recessives ” as they may be termed. In assuming the 
race, as before, to have reached the steady state, in the second 
generation after the first cross there will again be pure B’s, Bb- 
heterozygotes and pure b’s in the proportion of 1: 2: 1. But the pure 
B’s and the Z2fr-zygotes both develope 70 Q / 0 of ^-forms and 30°/ 0 of 
a’s; then’s develope 35°/ 0 of A -forms and 65 Q / 0 of a’s; therefore the 
proportion of ^4-forms in the race will be 
3 x 70 + 35 245 
= - - 61*25°/ 0 
4 4 
while of these A forms ff T) or 28-57% are pure, or 57-14 °/ 0 
hybrids, and or 14-29 °/o aberrant recessives. As before, let T n 
be the whole number of dominant forms, all of whose ancestors 
in one line are also dominants, in any one generation, and of these 
let pjj be pure forms, i n impure or hybrids, and e a aberrant recessives. 
Then 1 find that the following relations, corresponding to (i i) and 
(12), or (5) and (6) hold good :— 
Pn + 1 = *35 p n + -175 i n (14) 
i n + 1 = *35 p n + -35 i„ -fi -35 e n (15) 
e nH = -0875 i n + -175 e n (16) 
where of course / 
T« = Pn + in + e„ (17) 
These equations do not appear to give readily an expression 
relating the chance C n to C n . j as in equations (10) and (13), the 
aberrant recessives forming a troublesome element, but they enable 
the values of T 2 T 3 etc. to be calculated directly from the figures 
given above, say T x = 245, p l = 70, i t = 140, e x = 35 (the 
absolute figures do not matter). Then C n is given at once by the 
ratio of T n + 1 to T n . Using this process I find 
C 1 = -62500 
C 2 = -63000 
C t! = -63194 
G 4 = -63269 
C 5 = -63298 
