2 ^ 0 Mr. Morgan on Survivorships. 
three lives, nor are they even more simple, so that it can now 
be seldom necessary to have recourse to them. Without the 
assistance of the preceding lemma, and the computations which 
have been just made, it would not have been possible to have 
ascertained the degree of accuracy of any approximation ; and 
therefore were no other end answered by them, this of itself 
would be of sufficient consequence to deserve the tune and 
labour which I have bestowed upon this subject. But it will 
appear, in the solution of some of the succeeding problems, 
that the use and application of this lemma, and especially of 
the table deduced from it, are much more extensive and lm- 
portant. 
PROBLEM II. 
To find the value of an annuity during the life of C, after 
the decease of A, provided A should survive B. 
SOLUTION. 
The payment of this annuity depends only on one contin- 
gency ; and that is, the extinction of the two lives of A and 
B before the end of each year (B having died fifst), and the 
continuance of the life of C to the end of thcce respective 
years. The value therefore of the annuity for the first year 
will be = mi 4 for the second year = — --’ forthe 
third year and so 011 f ° r the remaining y ears ' 
The value of the annuity ;( when C is the oldest life) will conse- 
.da' , e . a' -j- a " t 
quently be expressed by the two series zacr *” 2 acr z 
