238 
Mr. Morgan on Survivorships. 
of the three lives, will be = — 7 h <P • ^ 7 — 
Thirdly. If B be the oldest of the three lives , let x denote the 
number of years between the ages of B and of the last person 
in the table, C' the value of an annuity on the life of C for x 
years, A'C' the same value on the joint lives of A and C, and ?r 
the probability (found by the table in the foregoing lemma), 
that B dies after A. Then, by proceeding as above, the value 
0 * A'C* 
of the annuity in this ease will be found = — b 7rx 
c — c" + BC ~ — . .Q. E.D. 
When the lives are all equal, the general rule deduced either 
C-CCC 
from the series or the foregoing expressions becomes = — - — ’ 
which is known to be the exact value in this case from self- 
evident principles. 
As this method of solution is applicable to a great number 
of problems, I have thought it necessary to make the following 
computations, with the view of determining how far it may 
be depended upon. It is to be observed, that the first series of 
b — m d | */-f 1" • b — n e y 
~2 ubcr t 2 abcr 1 
fractions in the above solution, or 
,&c.. should have been (according to the lemma), in order to 
. , a' . b — m . d , a' ■ b—n -f ' + '' ™ ~ n • e f 
express the exact value, - 1 2ubtr > T 
&c. and that it is impossible to find a general expression 
which shall be equal to this latter series and at the same time 
fit for use. This has rendered it necessary to have recourse 
to the present approximation. But in the first column of the 
following examples, each term of this last series has been se- 
parately computed, so that by comparing the values in that and 
the second column an exact idea may be formed of the accu- 
racy of the preceding rules. 
