SUCCESSIVE LIVES. 
187 
sion of a perpetuity of £i, being the perpetuity which might be pur 
chased for £l to be entered upon immediately. 
By the formula above, the present value of a reversion of a perpe 
tuity of £l is * — a m ; this multiplied by/, gives 1—ia m , the value 
of the reversion of £i per annum, as before. 
Example. What is the present value of a perpetuity of £50 per 
annum, to be entered upon after the failure of the existence of a life 
aged 39 ? (Carlisle 4 per cent.) 
.04)1.00 
25 
«39=15.2718 
9.7282 
50 
486.4KT= £486 8 2. 
238. An annuity is to be enjoyed during the existence of a life aged 
m, and at his decease a successor is to be named who is to enjoy the 
annuity daring his life. Required an expression for the present value 
of the annuity on the second life. 
Let the value of an annuity of £l on the second life at the time of 
entering on possession be denoted by V 2 , then at the end of the year in 
which the existence of the present life shall cease, his successor will be 
put in possession of £l, and of an annuity whose value is V 2 : the 
present value of what is to be enjoyed by the second life is therefore 
the present value of an insurance of (1 + Y 2 ) pounds, payable at the end 
of the year in which the existence of a life aged m shall fail, viz., 
^~^(1 + V 2 ) Art. 188.) 
1 *4- z 
If to this a m be added, we have a m + ~ (1 + V 2 ), the value of 
It® 
the two successive lives. 
239. If there be three lives, and we call the value of the annuity on 
the third life at the time of entering on possession Y 3 , we have 1+V 3 , 
the sum of which the third life enters on possession at the end of the 
year in which the second shall cease to exist. If we call the value of 
the two successive lives x, then -r- — x is the present value of a per 
petuity to be entered upon at the same time. Now, when a perpetuity 
and any other sum are deferred for the same time, the value of the 
perpetuity at the time of entering on possession is to its present value 
as the value of that sum at the time of possession is to its present 
value: 
,.11 , 1—ix ,, . 
1+ i : r“* ::1+v>: TH ( +v,) ’