72 Inheritance of the Duration of Life 
dies, then the wife, if alive, will be a;2 = a;, — 1'287 years from 61"004 years, the 
probable age at death of the array of such women. Hence the chance that the 
woman survives her husband if he dies at : 
V2 
1 r"" 
27r J X, 
1 /■* 
where b = - 17 004 and a, = 17-9290. 
If she does survive him, her expectation of life 
^27ri. 
dx 
V27r^x, 
Hence her expectation of widowhood, if the man dies at x^ expressed in years, is 
the product of these expressions, 
V27r 
1 
^ e-^m:i^.?)dx 
Now the man's chance of dying between x-^ and x-^ + dx^ 
\/27r 
V27r^a 
where a = - 15-717 and o-, = 15-2830. 
We now multiply the woman's expectation of widowhood if the man's death 
takes place at a particular age by the man's chance of dying at that age and 
integrate for all possible ages of the man at death. We thus find if be the 
expectation of widowhood of the wife : 
-lir J a V27r 
Here x^=x^ — /i, where A = 1-287 years. 
Now let us write the subject of integration in the numerator in the form 
or — {x, - cy +p = -\ + -A - + — • 
0-2 ^ 0-,^ (To- o-/ aJ 
