288 
Mr. Gompertz’s analysis applicable to the 
tion 3, that if x be put = 7r + t, and that if when t be not 
greater than p, be equal to — tM'„, and N „ +i = 
— £N # W sufficiently near, that we shall have 0 
*+P 
t t 
M, 
M X .N X = 
xN, 
M„ — M'„ t x (— N' t) = — p N'„ M, 
+ £. M'.N',= —pw. x M .-4 M'„ = _/>N'„ . M, +£ 
2 
= — (N ff — N ff+ £) M „ +p . And therefore if M x represent 
2 
the chance that a certain circumstance shall exist at the time#, 
and N* the chance that some other certain object shall exist at 
the time x ; then the chance that the second circumstance shall 
fail during the existence of the first between the time n — p 
(N w X M v + lp ). 
and m is sufficiently nearly = n ^_ 
m—p 
And the present value of one pound to be received at the 
first of the periods n, n-\-p , ?i-\-2p, &c. from the present time, 
which shall happen after such failure, provided it shall take 
place between the time n — p and;;?, will be according to 
hypothesis with sufficient proximity = n ^_p 
m — p 
9r 
m — p 
And this mode of investigation wil 
(r' + ^xN»— N„ +# 
(r’ +t x ¥t, +p x 
afford different 
modes of solution to all the Examples of Section 4. For 
instance : if this be applied to Example 8, Art. 7, Section 4, by 
