Miscellanea 
177 
Hence in order tbiat the ofiFspring population should be stable, it is needful that in the array of 
offspring for given parents : 
(«) ^■ = 72''^- 
(6) s'W= 2^2 {n/^T - I {^W; + -^'W')] = 2s/2 v//V' (l - l) = 
if /3i"' = /3j' = /3i", i.e. the skewness be the same for fathers, mothers and offspring. 
/33'^ = 2(7/32"'-15), 
if /32"' = ^2' = /3.;'. 
Thus, we have for the array of offspring of given parents 
1 \ 
' = 72 
/32'^-3=^(/V-3) 
Accordingly the variability of the array is less than that of the population of offspring ; and 
the array (unless /3i"'=0, /iji"' = 3) is more .skew and has greater kurtosis than the general 
population. 
If J'i2, >'23, ''31 be the three con-elations of father, mother and offspring we know that the mean 
standard-deviation of the offspring of arrays having the same parents is 
/l - ni--r^ - ri2^ + 2;-i2/-i3»-3i 
s=as^ YZr,,2 ' 
and this equals if there be no assortative mating 
('•12 = 0), 0-3^1- ria^-j-as^. 
If we could assume this opial to s we must have, since 
.(xiii). 
1 
V2 
leading to ^i.-j^ + j-^i^ = ^ , 
or if the two parental correlations are equal to 
n3 = »'23="5- 
In other words, if the i)arental influences were equal and there were no assortative mating 
and the character in the array of offspring had the mean value 
Z VcTi 0-2, 
then the population could only be stable if 
'•i3 = >V! = 0-5. 
But this apparentlj' noteworthy result only l)egs the question. By the general theory of 
correlation the mean of tlie array of offspring is 
z + ir. ( ■^Izl' + ^•23-''i2 ''i3 y-j/ 
fx— X y — y 
= « + 0-3( '•is + ''23'^ ■ 
\ 0-1 0-2 
if thei'e be no assortative mating. 
= 0^3 I ' i;i H ' 23 — ) + '■ - (T-i 1 . 
fl " T-i/ \ ''"l ""2 / 
Biometrika x 23 
