CONNECTED WITH HUMAN MOETALITY. 
545 
Art. 24. In applying the formula c^L^, the Napierian logarithm of the number of per- 
sons living at the age .r, 
to the useful purposes to which it is serviceable, it stands in need of reduction into 
another, for which might be represented an expression like 
&c. ; 
but such a form might not have the converging property requisite for the purpose 
required if x were a large number : but if a represent a certain age, and the age to which 
the law were meant to apply were represented by a-\-x instead of x, then the formula 
would stand _____ 
and would admit of the terms of each member to stand in the above form of x, x^, 
with given converging coefficients, and each of the members of the last equation would be 
convertible into a converging series for most or all the values which x would be required 
to have ; and therefore the right side of the last equation, which would consist of all the 
developments together, would form a series, as above, which would be convenient, and 
might be expressed by 
Ao-l-A,.r+A 2 ^^+A 3 a^®+ &c. ; 
and the Napierian logarithm of the number of persons living at the age a would be 
represented by the first term Ao of that series ; and the Napierian logarithm of 
that is, of the chance of a person now of the age a being living in x years, would be 
because is 
= Ai^ -f Aj^^ + Agt^® + &c. , 
= € X S*=€“x(i+ &V-1-&C.), 
2c4 2.3 cA 2.3.4c1 ’ 
because 6, the Napierian €, is very small, and a very few terms of the above series will 
be sufficient. Similarly, f will be 
f X (1 ~^x'-\^~J'\x\ &c.) ; 
and if x is equal to or greater than one, would be of perfectly insignificant value, and 
omissible. 
I may observe that 
£“+* = 2“X &c.) 
is a very convergent series ; and if a be equal to 20 in the Carlisle mortality, or above 
that value, would be of insignificant value, and the term which depends on it omissible ; 
in like manner the term depending on which is 
=v“x 
will be converging, and will be insignificant whilst a-^x is less than 80, but will be 
significant when a-\-x is greater than 80. 
4 E 2 
