Miscellanea 175 
III. Note on the essential Conditions that a Population breeding at 
random should be in a Stable State. 
By K. PEARSON, F.R.S. 
Let us deal with bi-parental inheritance in the first place. Let x be a character in the father, 
mean x, standard deviation o-j ; lot y be the same character in the mother, y its mean, and 0-2 its 
standard deviation. Let z be the character in offspring of one sex, 0-3 l)C the standard deviation of 
all offspring of this sex and i the mean. Let yi.^, ns, jx^ ; /^a", /x;/', ^14" ; and /x.^'", /X3'", /X4'", be the 
moment coefficients about the means respectively of father, mother and offspring frequency distri- 
butions. Let be the mean of the offspring of those parents, who have characters x and y, and 
let the array of frequency of such offspring be given by {u) du about i.e. the character of any 
offspring in this array is z^y + u, where u is independent of the parental characters x and y, but 
Zxy is a function of x and y the parental characters. Some writers have suggested that the 
offspring character should be taken as a blend of the parental characters, i.e. 
z = \{x+y\ 
understanding by blend the mean of the parental characters. This apjjears to be very unsatis- 
factory for : 
(a) It supposes the parental characters to fix absolutely the otfspi'ing characters which is far 
from a result of experience. 
{b) It supposes the mother to reproduce the female size of character in the male and the 
female offspring alike, whereas she contributes to each the sex character of her own stock, i.e. if 
she is a tall woman, she would contribute absolutely more to a son than to a daughter. The late 
Sir Francis Galton got over this difficulty by "reducing female measures to their male equiva- 
lents." This he did by altering absolute measurements in the ratio of male to female mean 
measurements. Thus he would take for the mean of his array of offspring 
z^y = ^[a; + ^y 
if he were dealing with male offspring. A more reasonable hypothesis is to assume that 
. = U^ + ^-) + u (i). 
This will practically agree with Sir' Francis's form, if the coefficients of \-ariation in the two sexes 
are the same, i.e. a-ilx = cr2/y. 
If we measure u from the mean of the array of offspring we have 
(ii). 
We shall now suppose the offspring to follow the law (i), or 
^=*^3(^ + «)+« (iii), 
where x and y are uncorrelated (mating at random), and u represents other influences than the 
parental, and is therefore uncorrelated with x and y*. The frequency distributions of x and y 
* This assumes the homoscedastieity of the arrays of offspring clue to pairs of fathers and mothers 
with characters x and y. 
