236 Mr. Morgan on Survivorships. 
with another contingency, which necessarily renders them of 
less consequence. The solution, therefore, particularly in the 
former case, becomes very easy ; and even in the latter, by the 
assistance of the table in my first paper,* it becomes Equally 
simple and correct. But when B is the oldest of the three 
lives, the above fractions are combined with a series which is 
often of considerable importance, and consequently the com- 
mon method of solution fails in this case. Yet even here, be- 
ing possessed of the table deduced from the foregoing lemma, 
it is attended with little or no difficulty, and a general rule 
as short and accurate is obtained as in the other cases. This 
however will be more satisfactorily proved by the following 
operations. 
ist. Let C be the oldest of the three lives. In the first year 
the payment of the annuity depends on one or other of two 
events ; either that A and B both die (B having died last), and 
that C lives, the probability of which event is expressed by 
a ' . b~Hi . d or t | iat on ]y a dies, a nd that B and C both live, which 
zabc ’ 
probability is expressed by The value, therefore, of the 
annuity for the first year will be = In the second 
year, the payment of the annuity depends nearly on the same 
events: ist. that A and B both die in the first or second year 
(B having died last), and that C lives to the end of this term, 
which is = -L or 2dly, that only A has died before 
the end of the second year, and that B and C have both lived, 
which is = Hence the value of the annuity for the 
• Phil. Trans. Vol. LXXVIII. p. 337- 
