224 Mr. Morgan on Survivorships. 
after another in a given time. It becomes necessary, therefore, 
previous to any other investigation, to deduce a general me- 
thod of ascertaining such an event, and for this purpose I shall 
subjoin the following lemma. 
LEMMA. 
To determine, from any table of observations, the probabi- 
lity that B the elder dies after A the younger of two lives, 
either in any given number of years, or during the whole con- 
tinuance of the life of B. 
SOLUTION. 
This event can take place in the first year only by the ex- 
tinction of both lives, A having died first ; the probability of 
which will be expressed by the fraction b * In the se- 
cond year the probability will be increased ; for the event 
may have taken place, as above mentioned, in the first year, or 
the lives may have failed in the second year, A having died 
first ; or B may have died in this year, and A in the first 
year. The expression, therefore, for the second year will be 
— n a -v a" 
h — m 
+ 
+ 
i b 
—r X — 
+ 
. In 
2 ab 1 zab 1 ab 
the third year the probability will be still further increased ; 
for, in addition to the foregoing contingencies, the event may 
have taken place by the extinction of the two lives in the third 
year, A having died first ; or by the extinction of the life of A 
* In order to avoid unnecessary repetitions, I have uniformly in this paper preserved 
the same symbols as in my last paper. -See PhiL Trans. Vol. LXXXI. page *47. 
