estimation of the value of life contingencies. 
291 
new series, whereof the first term is equal to the 1st -j- 9th -f- 
17th &c. term of the original series; the second term equal 
to 2nd -j- 10th + &c. term of the original series, &c. it will 
follow if our original series be a gradually, but a very slowly 
A 
converging series, such that —^differs very little from unity ; 
A 1 / 
7T-\- I ' 
since each term of the new series will be nearly equal to each 
other ; that if this method be used, we shall have a means of 
detecting any great error, as it would be evinced by a too great 
difference produced in different terms. And if an approxi- 
mation to the value be sufficient, we may avoid great labour 
by taking 8 x 
K, or 8 x l 
78 
A w for the value of the sum; 
and if r does not differ much from unity 
= 8 . 
-i— r 9 8 
7=T X 5 
77 
r* + A„ or to 8 . 
I — r 9 8 
1 — r * 6 
78 
A ff r will be nearly 
r *~ 5 A_. 
Art. 5. It may also be serviceable to observe, that if M*., 
M x M) by 
M' , M" , do not contain x or y, that ( M" . 0 
*1 y 
x y 
M" N = M" N" — 
y y x » 
putting ° n 
M y M = N will become « 
x x y n 
x 
M'bN / — °, 
n n n ' 
M N = M" N 
y y XX n n n ' 
M" M M = 
y y y 
y 
y 
y 
*]m' m - 
- M", x °„ 
M' M — 0 
1 
1 
i 
nr n 
y y n 
X 
n' 
w 
M" M' M ; because 
y y y 
