ENDOWMENTS. 
The present value of £l to he received at the end of n years, pro 
vided a life now aged rn, survive that term, is- 
lm-i-n if 
the annual premium for the same, payable n years at the beginning of 
each year is 
N^-N, 
The value of £l to be received at the end of the year in which a life 
aged m shall die, provided that event happen within n years, or to be 
received at the end of n years if the life survive that time is 
M,„ _ l _ U?n-fn ' -^dm+n 
The annual premium is 
M m -f" —* lVX m ^. n 
N ZIn ’ 
x ' m—1 ^ m+n -1 
REVERSIONARY ANNUITIES. 
On One Life after the failure of another. 
The single premium for an annuity on a life aged m after the failure 
of another aged m x , is 
O'm ^m, vi\ 9 
the annual premium for the same is 
On One Life after the failure of the Joint Existence of two others. 
The value of an annuity on a life aged m after the failure of the joint 
existence of two others aged m x and m. 2 , is 
a m n, m\, m 2 > 
the annual premium for the same is 
f “1” V/i, m\, ill.> 
On One Life after the decease of the last Survivor of two others. 
The value of an annuity on a life aged m, after the death of the last 
survivor of two others aged m, and m 2 , is 
O'U ei rn . nr[ mi, «¡2 > 
the annual premium for the same is