296 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
parentage, or influenced by heredity, being general statistics, we shall assume that, 
on the average, A 2 = k L, and hence : 
. k k (k~ erj) a , [ ki 7 (17 + e) ] cr . 
Ai — —z - 1 -5—r(J-i — yk) — 1 K + ; .v , \ v(h — y]L 2 ). 
1 — rj~ — K' a 1 1 — r]- — k. ] a ' 
Similarly : 
t = u + 7 
These formulas give the chief influence of age of appearance and intensity of 
disease in parent upon intensity and age of appearance in the offspring. If we 
suppose k positive, i.e., if increased age of appearance means for the diseased 
population as a whole increased intensity, then intensity of disease in parents tends 
to lower the age at which the disease appears in the offspring, and this tendency to 
antedate is the greater, the greater the correlation (77) between intensity of the 
disease in parent and child, i.e., the stronger the hereditability of the disease. If k 
be negative, i.e., increased age of appearance means for the diseased population as a 
whole decreased intensity, then the opposite result will follow, for will have a less 
negative value than if 77 = 0 , i.e., the age of offspring be raised towards the mean.* 
It would thus seem possible that the antedating of inheritance in the case of gout 
and diabetes might correspond to a post-dating in the cases of diseases intenser hi 
youthful incidence. 
Our second formula shows that for diseases with increased intensity at increased 
age of appearance, a late age of appearance in the parent decreases the intensity of 
appearance in the offspring, while the reverse holds if the disease is intenser for 
youthful than for senile incidence. 
It must be noted that the correlation between intensity and age without regard to 
heredity is given by : 
Ii = * - a 15 
G 
so that heredity affects the constant of correlation k by multiplying it by the 
quantity : 
, e(?; + e) 
The second part of this expression is by no means necessarily negligible as 
compared with the first part, if heredity be strong. For example, with the order of 
correlation we have found between parent and offspring, in the case of stature the 
* Generally but not absolutely, for y~ + G for some diseases may be > 1, and, if not very different, 
then the second term is the important term 
