PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 295 
(c.) Antedating of Family Diseases .—We may now apply these results to the case 
of morbid inheritance, making the following assumptions :— 
(a.) The distribution of the disease with regard to both age and intensity will be 
taken to be the same for any two successive generations, and to be normal. 
(b.) The age at which the disease appears and its intensity are both directly 
inherited, but the age of appearance and intensity of the disease in the parent are 
not directly correlated with the intensity of the disease and the age of its reappear¬ 
ance in the offspring. 
Let e be the coefficient of correlation between the age of appearance of disease in 
the parent and the age of the offspring at its reappearance ; let a be the standard- 
deviation for the frequency of the disease at different ages, and M the mean age at 
which the disease appears in the population. 
Let y be the coefficient of correlation between the intensity of the disease in the 
parent and the intensity of the disease in the offspring; let cr' the standard-deviation 
of the intensity-frequency and M' be the mean intensity.'"' 
Let M + Aj, M' + I l9 be the mean age of the appearance of the disease and its 
mean intensity for an array of offspring, whose parents exhibited the disease 
when M + A 2 years old with an intensity M' + I 2 . 
Let the subscripts 1, 2, 3, 4 refer respectively to age of offspring, age of parent, 
and intensity in offspring and intensity in parent. Then, in the formula for triple 
correlation, we must put: 
r l2 = e > r 3i — V’ r l3 = = K > r l4 = r 23 = 0- 
Hence: 
— l — f — k 2 , v 12 = e + y (k 3 — eg), 
\ 3 = X 4 = 1 ~ e 2 — k 3 , = y + e ( k 2 — ey), 
v i 3 = v u = k — k(k 2 — ey), 
v n = L>3 = — «(e + y), 
1/X = 1 — e 2 — y 2 — 2k 2 -f (k 3 — eyf. 
Substituting these values in the regression formula, we find : 
\ e + v ~ eq) 
-^-1 — -i 
1-7? 
d — 67?) A ( K — K. (k~ — 67?) a T K (e + 7?) (J y 
o O -^*2 I 1 q O / -^1 1 2 O / -*-2* 
" — 1 — Tj" — a I — 7) — K a 
Now as the parents in the group M + A 2 , M' + I 2 are in no way selected by 
* It might be difficult to get a mathematical measure of the intensity of a disease. For simple 
theory as apart from statistical measurements, such is, however, unnecessary. The terms used in 
medical works, acute, subacute, chronic, &c., sufficiently indicate that the relative intensity of various 
cases is a fact duly recognized by the trained medical mind, if it cannot always be quantitatively 
expressed. 
