9.6 o Mr. Gompertz’s analysis applicable to the 
continuance of K" ; and provided also, it happens between the 
time n — -p and m ; the present value of one pound certain to 
be received in one year being equal to r. 
Article 6. As it is probable that the reader will be desirous 
to see the application of our theory to some of the problems 
which have been esteemed the most difficult, I shall consider 
the problems which have been solved by Mr. Morgan, in 
the Philosophical Transactions ; and in pursuing this object, 
I shall take the examples in the order in which they have 
been presented by Mr. Baily, in his work on Assurances, 
from page 9.06, because that Gentleman’s book was in my 
hand when I worked the examples. Note, that the object 
which I denote by p , has with those Gentlemen the par- 
ticular value one year ; and is supposed by them to be 
wholly within the limits of proportional decrement. Note 
also, that the present ages of A, B and C, I denote by a, 
b and c. 
Example 1. The chance of A dying, the first of the three 
lives A, B, C, is by taking K" of No. 2, in the last article = 
— ^ b ’ c , and K"'= — , the fluent of J — L a+X - 
b, c « a, b, c 
If p be within the limits of proportional decrements (that 
is of decrements proportionate to the times) then by Section 3, 
Article 9, the chance of the events happening between the 
L x «i 
0 
7 r+/>, that is » 
v+P t 
L . 
x:b,c 
^ a, bj c 
12 L 
a 
X ' 
L , — L . . 
a+re a+ir+p 
IS L— * 
a,b,c 
L c+v~~ L c+v+p 
or if p be small ; such for instance as one year, it will be 
