CONNECTED WITH HUMAN MORTALITY. 
549 
expression of a logarithm of an analytical expression, and as there are or may be in 
some applications of the theorems many entanglements of logarithms in analytical 
expressions, and anti-analytical expressions of logarithms, I will now give different 
methods of operation for those purposes. I will commence by stating that if the ana- 
lytical logarithmic expression is that of the common logarithm, then this must be con- 
verted to that of the Napierian logarithm by the multiplication of the coefficients by the 
Napierian logarithm of 10, that is, by the reciprocal of *4342944, &c. ; but a very few 
terms of this decimal will be required. Let the Napierian logarithm be represented by 
Ao+Aia7+A2^*-4-A3a^^+ &c.. 
and by the notation I have adopted because <=-Aa will express the anti-Napierian 
logarithm of Aq, that is to say, the natural number of which A, is the anti-Napierian 
logarithm, and the expression, namely the anti-Napierian logarithm of 
will stand 
hut 
(Ao+Aia:+A 2 a;"+ &c.), 
<=-Ao X <^(Ai^+ Aa^^d- A3ic®+ &c.) ; 
Ai^; + Aj^^ + As^® + &c. ), 
or the anti-Napierian logarithm of 
&c.), 
is 
= 1 +Aia'-|-A 2 ar^- 1 - &c. &c.j +^Ai^-|- &c.f &c. 
— 1 “h Aj^ A^oif “h A^oif -{- A^x* 
-h 
I &c. ; 
+AA;ir’+|A;A^‘ 
-I — ^AV^ 
^ 2.3.4 
/ 
and consequently if the anti-Napierian logarithm of Ao+Ai^+Aa^^'-f &c. be expressed 
’oy &c., we shall have the following equations between Bo, Bj, B^, 
&c., and Ao, Aj, Aj, &c. : 
Bo=c-Ao; B,=BoA,; B2=Bo(A2+iA:) ; 
B 3 BoX ^Aa-f-AiAaffi g^^Ai^ ; B^ — Bo^A4-l-A,A2+-A2+2AiAa-f-2_3 4 -^.^^, &c. ; 
so that as Bq, Bj, Bj, &c. are found by these equations from Aj, Aj, A 3 , &c., and on the 
contrary, Aq, Aj, Aj, &c. are found from Bo, B,, Bj, &c. from the equations 
-^0 ^Bo, Aj g X Bj, Aj T> X B, 9 Aj ; 
aJq -Dq 
AjAj— j^Aj,* A^rr^x B^— AjAj— 
^A^-^A^A 
1 
* 2.3.4 
A\, «&c. ; 
