REVERSIONS. 
153 
and the value of 2 q {m 0 „being variable during the possible term of 
"1 (0 
the joint existence of B and C, there is no other general and certain 
method of calculating the present values of the annual payments during 
that period, than to calculate the value of each year’s payment sepa 
rately, and add the whole together. 
Let the number of years between the age of the older of the lives B 
and C, and the limiting age in the table be denoted by z, then the pre 
sent value of the annual payments to be received after that period will be 
^ i.mj,) n X Cl m • 
(0 I s 
The payment of the annuity during any of the first z years, as the 
nth, depends on the following events ; first, that A and C shall be 
living, and B dead, the probability of which is p (m , m ^, n —p im ,; 
second, that A shall be living, and B and C both dead, B having died 
first. Let y be assumed as the constant probability during the first 
z years, that provided B and C be both dead, B shall have died first 
of the two, then the probability of the second event is 
7/(1 Pm^n) (1 —Pw 2 , n ) Pm, n '—yiPm,n P(m,m{), n ~ P (m, m 2 ),n'f ^(m, m 2 , n )j 
adding to this the probability of the first event, we have 
P(m, m 2 ),n P(m,m 1 ,m 2 '),n~hy(p m,n P(rn,m{),n P(m.m z ),n~{'P(m,m l ,m ;l ),n')‘ 
185. If the annual decrement be supposed to be constant during the 
term for each of the lives, then y will become and the expression 
will become 
¥,P m, n" ~P(.m,mi), n “hP (m, w? 2 ), n P (m, n>l> I»2), n) > 
therefore the value of the annuity for the first z years is 
® d - ®{m, ®(«i, mj’ m 2) ) > 
*1 *1 *1 *1 
to which if we add 2y (mi w ) „ x a m , we have the total value, 
(!) ’ T 
mi) d" *“ mg) mo), n 
*1 - *1 0) i*