Miscellanea 
239 
V. Further Remarks on the Law of Ancestral Heredity*. 
By KARL PEARSON, F.R.S. 
This law is a rule for predicting the average value of a character in the offspring of ancestry 
with given characters. 
It is based on the following assumptions : 
(i) The proper prediction formula is the multiple regression formula. 
(ii) The individual ancestors in any generation may be replaced by the mid-ancestor of this 
generation, i.e. an individual supposed to have, as his character, that linear function of the 
characters of all the ancestors in that generation which is most highly correlated with the 
character of the offspring. 
(iii) The individual ancestral correlations decrease in geometrical progression. This is 
very close to the truth in practically all the races that have been statistically investigated. 
The general multiple regression formulae are 
x 0 -x 0 =S ^c p ^(X p -X p )j (i), 
2 0 =<to s/ 1 ~ $ ( c p r po) (ii), 
where X p is the character of the pth mid-ancestor, r po the correlation of this ancestor and the 
offspring, while 
( Hi )> 
r p _,=2*(? -.9) p^jjQ + €p ) (l+ fq ) (iv), 
where e p is the assortative mating in the pth generation, <r p is the variability of that generation 
and p p - q = average correlation of the individual pth ancestor with the offspring of the ^th 
generation. 
Further we have to determine the c's 
.(v). 
>'01 = Ci + C 2 ri2 + C 3 n3+ ••• + C » r l» 
r 03 = c 1 r 21 + c 2 +c 3 /- 2 3+ ... +c n r 2ll 
>'03=Cl? - 31 + C 2 > 1 32 + '"3+ ••• +V3» 
Assume r pq = afi»~ q , which will clearly be the case if p p _ g is of the form 
(8 p - q (vi), 
or in accord with our condition (iii). We shall show that c p = yrj" satisfies the above equations. 
Substituting we have 
af3 = y(r 1 + T,ZaP + T } 3 ap 3 + ...\ 
ap = y( W P + riZ + r ! lal3+ ... I ... 
ap = y(r ] ci(i'> + r t la(3 + r l l+ ... f 
* Reproduced from lecture notes on multiple regression. 
