140 .LIFE ANNUITIES. 
160. If the annuity cease at the expiration of t years from the present 
time the present value will be 
®(m) a Cn>,m,) 
n n • 
161. To find the value of the annuity payable during the joint lives of 
A and B respectively, aged m and and also during t years after the 
death of B, provided A shall live so long. 
The value of the annuity during the two joint lives is a,„, mi . 
The remaining part consists of two portions, one during the next 
t years, the value of which, by the last Article, is a m — a (m> _ ) , and 
«1 _ ‘1 
the other after t years; the value of any payment of which, as the ??th, 
will be the present value of £l, due at the end of n years, multiplied by 
the chance of A surviving that period, and of B having died within t 
years of that time, viz.— 
T • Pm,n (,Pm 1 ,n—t P m l i") — ? (-Pm,n • Vm y ,n—t P(m,m l ), n ) 
in which, p m ,n.p mi ,n-t 
¿m+n 
ilm+n Cnj+n—t \ Ci t —t P(m, wii—Q,n 
Ci \ Ci C’-i—t / Pt 
the expression therefore becomes r n (^ 0,w —P(m, m ^,n )> 
\ Pwi—i, t / 
the successive values of which, being found for every year after the ¿th, 
will give for their sum 
m, mx-0 
n 
Pmi—t,t 
a (.m, mi) > 
If 
adding to which the value of the other portions, we obtain for the total 
value required 
5f m,mi't' a (m) a (m,mj) + ( ’ 1 () - 
n n 
Pm L —l, i 
3y Davies’s Tables,— • 
1 ^(m, rni-i). 
^(m, mi) 1 — ®‘On)”j 
n# in 
N m+ , 
»(«) nV 
Vhen m is greater than m l — t, 
71 1 «1-0 I mi—t bijn+i, mi 
]< 
*mi—t, t D. 
m, mi—t 
Ci i f’m vni—i' 
N 
m+t, 
N 
m-H, w j