70 
Inheritance of the Duration of Life 
Turning now to the case of A himself, we shall use his parents' probable lives, 
87'935 and 85"388 years, as if they were their actual ages at death*, but what 
class are we to suppose him to belong to ? Does he belong to the mean popula- 
tion of adult sons ? If so his class's mean age is 53'490 years. Actually he is a 
father, without having lost a child, or without any of his children reaching the age 
of 21. Hence all we can do is to class him under the general group of fathers with 
a mean age of 66"384 yeai-s. Thus we have 
A'fi probable age at death 
= 66-334 + -1802 (87-935 - 68-370) + -1486 (85-388 - 67-947) 
= 72-748 years. 
Now the mean s. D. of all fathers = 14-8472, and therefore the s.D. of fathers 
with selected parents = 14-5810. 
Thus A belongs to an array of men who die at the average age of 72-748 years 
with a standard deviation of 14-5810 years. 
* This assumption is not strictly legitimate for the reasons given in the next illustration. If the 
mean age of death of fathers be )», and of mothers and the father die at m^ + x and the mother at 
m^ + y, the probable age at death of the son would be found from an equation of the form given above 
in Table B to be : m.j = c^ + c.,x + c^y . Hence the expectation of life y oi & man aged vi^-a^ whose 
parents die at Wj + x and )«., + y would be 
V27r I \l2ir J -(m^-a-s) 
Now if the parents be alive at oij + Oi and vu + a.^ years their chances of dying between m, + a; and 
m.2 + y and m-^ + x + Sx, m.^ + y + Sy will be 
1 _^ / 1 1 _Jl / 1 /•« 
e ^<ri' dx / -— I e 2<r,!i dx and —7= e ^"21 dy / I e 2<rj2 dy 
respectively. 
Multiply the product of these chances by the expectation of life E^^ y ; we have on integrating for 
x and y from to x and from a., to oo respectively, the total value E of the man's expectation of 
life : 
/ e 2o-,2 dx X I e 2(r,^ dy 
s/2irjai 'J'2irjai 
Here is the function of x and y given above, S is the standard deviation of sons of selected 
parents, tr, and a.^ the standard deviations of fathers and mothers of adult sons. Without using 
troublesome quadrature formulae I do not at present see how to evaluate the integral in the numerator. 
It certainly deserves investigation. 
