Mr. Gompertz’s analysis applicable to the 
2 2 6 
in the same manner n— p 
m — p 
p is the chance of its not 
having failed on or before the time 7r — p ; and the excess of 
the latter above the former, is the chance of its failing be- 
tween the intervals 7r — p and tt, which multiplied by r v gives 
for the present value of one pound, to be 
P \ j. P 
v :—p\ x r* nr 
vr—p\ C nr 
received at the expiration of the time tt, in case the condition 
should fail between the time tt — p and % ; and if *r be succes- 
sively interpreted by n, n-j-p ; n-^-Qp, &c. m, the sum of the re- 
sulting expressions, will be the value of one pound to be re- 
ceived at the first of the periods/) from the time n — p that shall 
happen after the failure of the condition, provided that that 
failure takes place between the periods n — p and m ; but because 
V ' 
P 
P 
- — Hh n +p 
' _ 1 _ P 
-f- n + 2p 
n P 
-f- &C m 
C m 
C m+p 
C n + 2y 
c m 
and 
similarly n—p 
P 
, P 
— — -J- n 
, P 
— n+p 
m — p 
C n — p 
C n 
C n+p 
— . -J- &c. 
c 
p 
m — p 
m—p 
T 
V- 
P 
n 
m 
r 
T 
; therefore the said value is equal to n —p 
C m — p 
as above asserted. 
x r f — 
From the two equations first cited, it appears that n—p 
m — p 
r r r 
i P 
T n ~P 
n—p 
; the value of the above contingency 
