242 
Miscellanea 
Application IV. Return now to the case of a continuous variable for which the law of 
ancestral heredity was first developed. Observation seems to show that we have nearly 
if> lb "J x (:j) 2 > e ^ c - 
Hence - r p _ (Z = (\/2)P-«|x (§) p -«, 
3 
= -9428, 
whence rf- 2-12134t/ + 1 =0, 
, = •70712 = ^, ^ = -2357, 
y=-3333, v^ 2 = l'0000. 
Thus the series becomes very nearly 
#o - #o = i ( A", - X\ + X 2 - X 2 + X 3 -X 3 + ...). 
This result is noteworthy ; it shows that with the values found, three generations would 
suffice to produce offspring with the full selected character. 
Here every individual ancestor provides on the average half the contribution of the ancestor 
one grade closer and not one quarter as in Galton's Rule. Heuce by continual selection, we could 
advance a character beyond the selected value. How far such a selection is possible we do not 
know. The correlation with other characters would probably introduce counteracting selection. 
We have in this case 
2 0 = (T 0 Vl^ft 2 = <t 0 Jl- (-5773)2, 
whence the correlation for brother= - 5773, and 
2 0 =tr ( >x -8165, 
or we have an 18 °/ 0 reduction in variability by in-breeding. 
Galton's Conditio?!. 
Galton made it a condition of his series that if all the ancestry were selected and of value 
the offspring should have the same deviation 
In our case x 0 — x 0 = y{q^2(X 1 - X 1 ) + (rj\/2) 2 (X 2 - X 2 ) + etc.} 
='y7V2A/(l-jjV2), 
if all the ancestry deviate by h. Hence 
yr; V2/(l -t;n/2) = 1. 
This is the generalised Galton condition. 
It is very doubtful, however, whether it ever holds. Supposing it to hold, we may select for 
n generations only, then 
X-X 0 = y{r) J2h + (r)\/2y 2 h+ ... + (r,s/2)"h}, 
if the other mid-parents were non-selected. 
Hence the offspring h h — — - —. — — - — , 
L ° l—Tjs/2 
h' = h {l-(i ? V2)»}, 
by the condition. Thus if 17 V 2 differ at all from unity, we rapidly get the full effect of selection, 
i.e. offspring =/<. If, however, we stop selection at the wth generation, we have for the offspring 
of this generation with in-breeding 
x - x 0 =y [v J2h {1 - (r) V 2)"} + (t) V 2) 2 h + (rj V 2) 3 h + (r, V 2)" + 1 It] 
=y\ n v / 2/i + ('?\ / 2) 2 A + ( 7 V2) 3 / t -|- ... +( 7 V2)»A} 
or, if selection be stopped at any stage, there will on in-breeding be no regression. 
