CONNECTED WITH HUMAN MOETALITY. 
547 
there will only be in the values of ’Aq, ‘Aa, &c. the terms affected with € and e ; 
and when a is as great as 80, there will be only the terms affected with S, and v, as 
those affected with and s will be of total insignificance. And meanwhile it will be 
easy to have a Table calculated for every age from birth, even including the first 
months after birth, which will give all the values of *Ao, ‘A,, 'Ag, ‘Aj by bare inspec- 
tion. And to illustrate this observation, I will give the value of according to 
the Carlisle mortality, for the respective values of a, 30, 40, 50, as follows : — 
-=_M3o+,=2-17621- 
( -014087^ 
-h -0003357^=* 
^ + -000 
•0033924^* 
^ ® 95944^=* 
&c. 
-a-M3o+.=l-73924- { -030745^; 
+ -00050696.^" 
' -f @26777^* 
,-f@ 1395^^ 
&c. 
-^M,o,,=l-99926- 
f •021562^' 
+ •00041513^'^ 
' + (5) 2839^^ 
@122450?^ 
&c. 
Art. 25. But from what has been stated, the result of the analysis of any problem may 
give the present value due the irth year’s payment by an expression of the form of 
See., 
where A,,, Aj, Ag will be determined by the methods explained ^by the conditions of the 
problem. If we wish to have the present value of the sum due to every yearly payment 
between the values m and n of x, we have, according to our notation, to find the sum of 
X 
X 
X 
X 
11 1 
11 1 
1 
• K . k 'fn. 
tne series Ao »1^^-1-A, . « 
..J-l 1 A ^ . A m 
XP -j-Ag. “bAj. n 
xP'^‘^, where n 
signifies, when applied to 
J) __32 
the term x for instance, the sum m'’+w-{-l| -j-m-l-2j up to inclusively of m and n; 
and I therefore give a Table (see p. 543,) which I call a Collecting Table of Powers, to 
facilitate the summation. 
This Table for the collection of powers given does not go to the extent which may be 
required in complicated cases, though it may be sufficient, for instance, for finding the 
value of an annuity on several joint lives, or incases not very much more intricate; but 
when the powers required of x are much higher than this Table comprehends, it will 
not be difiicult, but, on the contrary, very easy to find the sum of these high powers of 
the series 1, 2, 3, &c. from the well-known theorem that 
xp=Ipj^2p-\-Zp-\- See. ^^by 
