estimation of the value of life contingencies. 227 
therefore becomes (r? — 1) . « 
r' 
7 
c 
r 
T 
^ p | 
-j- n—p m — — > . r 
L n—p C nt I C j 
X r p n 
m 
we shall 
If we look to the expression n —p- 
m — p 
discover that if the increment of some variable quantity v. be 
equal to p, and that be the only variable quantity concerned 
r 
in the functions — x r*, that our quantity, the value of 
m—y. C 
the assurance, will the increment thereof after k be made equal 
r 
p 
to 0, and the assurance, or the increment of m — * 
j* 
x r N 
thus modified, I shall denote by n 
m 
. And in conformity 
with this notation, I should write n 
m 
a, b, c, &c. for the assur- 
ance of one pound to be received at the first of the equal 
periods p from the time n — p which shall happen after the 
time n—p , and not after the expiration of the time m ; after 
the extinction of the joint lives of the persons now aged a , b , c, 
&c. also by n 
m 
a, b, C} &c. j should denote the present value 
of the assurance on similar conditions with respect to the 
failure of the last v survivors, &c. When m is infinite, or is 
supposed to be the greatest possible limit, we, in conformity 
with the notation hitherto used, would write those expres- 
