546 
ME. B. GOMPEETZ ON THE SCIENCE 
It remains now to consider the important term X a+a’— which for abbre- 
viation I call and putting ]i—a=.w, we get 
Ax &c.^ j 
= cr-|e“.§'„x ^—w—w.ex—^w.e] 
e\x*, &c. 
consequently, if be put =<= — e“$'oW, that is, equal to the anti-Napierian logarithm of 
—e'^.q^w, and we put 
Ai=e“5'oXl— we; 
A2=e“. §-0 X 1 - 9 we X e ; 
1 \ -;2 
.— gWe i.e| ; 
1 1-^3 
A,= ~e'^.q,.l—jwe.e\ ; &c.. 
-‘^3 2 ^ • S’o X ^1' 
we shall have 
— e“. g-oW -f A,^ 4- -f + &c. } 
=V„X'=-{Ai^4-A2^^4-A3a;®4-&c.} ; 
or making use of the theorem for finding the anti-Napierian logarithm above given, and 
taking Ao = 0, we have 
Ma+^=VaX(l4“B,^4“B2^^4“®3'^*4~ &c.); 
and consequently we have, for instance, in the Carlisle mortality, the Napierian loga- 
rithm of 
La+;.= 'Ao4-*A,a;4-'A2^^4-'A3.r®, &c., 
if 
>Ao=Cg“4-,A/4-As“- V„4-K ; 
* Aj = CS**. S 4- 4“ As“. £ — V^B, -j-iW'f'"- »' ; 
‘^3=^ ^2“.^ — Vo4“/^‘''‘^ ; 
*A4=&c. A5=&c. &c.. 
But here it is to be observed, that if a be equal to one year or more, the term afiected 
with ^ is insignificant ; and whilst a-\-x is less than 80, in the Carlisle mortality, for 
instance, the term afiected with v is insignificant ; and when a is as great as 20, the term 
afiected with z is insignificant ; which observation shows that though the values of *Ao, 
*Ai, ^Aa appear intricate, only some of the terms for any value of a come into play in 
them, and that the values have much more simplicity than is apparent, without taking 
that circumstance into consideration. And, for instance, when a-\-x is between 20 and 
80, or even between about 10 and 80, a being no tless than 10, in the Carlisle Table, 
