Multiplying numerator and denominator by r m : 
+ r m+2 .d m+2 +r m+ *. d m+3 + &c.+&c. 
L r m 
In Tables 11 to 16, the numerator of this fraction is given for every 
age in column M, and the denominator in column D : 
• 4 - 
m 1) ’ 
190. When columns D and N are given without the column M, the 
value of the assurance may be found by means of them alone, without 
previously calculating the value of the annuity, thus: 
Art. 187. A m —r+ra m — a m , adding and subtracting unity which 
does not alter the value, it becomes 
A m =1 — 1 +r+ra m -cf m =l -(1—r)(l + a m ); 
and since 
but by the construction of the tables D m +N m —N m _ n 
N m _ x 
A m =l — (l~r) 
D„ 
191. To find the annual premium. 
The first payment is usually made at the time of effecting the insur 
ance, and the subsequent premiums paid at the end of every year during 
the term of the assurance; the single premium, which is equivalent to 
the payment of an annual premium of £l, is evidently 
1 + a . 
(in, mi, m2, &c.) 
The following simple rule of proportion determines the annual premium: 
“As unity added to the present valve of £l per annum on the given 
life or lives, is to £l, so is the single premium required, to its equiva 
lent annual premiumor, in other words, divide the single premium 
required to insure the given sum by the present value of £l per annum 
on the given life or lives increased by unity : 
1 - (1 -Q (1 + c (Wt W1 . _ 
1 +a 
(m, «p m2> &c.) 
1 -f- a. 
-(1-0» 
(m, m L m2, &c.) 
when there is only one life it becomes 
1 , 
(1—0. 
By Davies’s method 
1 
l + a„ 
1 
D„ 
1 +o-„ 
1 + 
N. 
Dl 
D m +N„ 
N r