ASSURANCES. 
213 
On Two Joint Lives after the decease of a third. 
The value of an annuity on two joint lives aged in and m l} after the 
decease of a third aged m 2 , is 
mi u rn, mi, m 2 3 
the annual premium for the same is 
dm, mi dm, m\, m2 
* + d m , mi, m2 
On the Survivor of two Lives after the failure of a third. 
The value of an annuity on the survivor of two Jives aged m and m x , 
after the death of another aged m 2 , is 
d„ lL a m> mj ” d m , mjs flmj, m2 mi, m2 ) 
the annual premium for the same is 
a m-T a my d m , mi” d m , m2 ®mi, m2T" d W) mi , m% 
1 T d m , m 2 + dm m-z d m , m x , mo 
ASSURANCES. 
The single premium for an assurance on a single life aged m, is 
1 — ia„, 
A m -r—{l-r)a m , or 1 — (1— 0(1 + 0, or 
M m , . N m _, 
or TT* or 1 ” o ~o TT~ ; 
l+z * 
the annual premium for the same is 
A m 1Î , M 
or (i —r), or 
D„ 
1 + a„ 
1 +a v 
N , 5 N 
’ m—l J-’ m—1 
(1~0. 
The single ‘premium for an assurance on two joint lives aged m and 
m l is 
1 — id 
Jm.Trrr— (1 or 1—(l-r)(l + a„„ mi ), or - * 1 * 
1 m,vni — ' ‘ /^m, mi : 
the annual premium for the same is 
1-H 
A, 
1 -\-d n 
or 
1 T" d-m, : 
(1-0. 
The single premium for an assurance on the longest of two lives 
aged m and vn x , is 
-1=r— (1 — r)(a m + a mi - a m , 
or 
’ (m, mj) ' V-- ■ -"mi "m. 
1 — (1 - 0(1 + «m+a mi - a m , mj, 
1 — ¿( a m+<y—a m .w x ). 
l + i 1