﻿Exponential and Logarithmic Theorems. 37 



a 5 \ V 2 I 11 \ v 3 I 3a 4 



+ 1.2.3.4 ( a * -& c 0+& c - ( 2 )- If we substitute the value of y, 



in the second member of (2), then since the first member of the 

 equation is independent of a, and a is to be arbitrary, the se- 

 cond member of the equation must also be independent of a, 



hence we must have log.(l+a)=»»[a-^-+^-- j+&c), (3), 



and a 3 - fl . + j2« 4 - &c. = \'~— 1 } ,« 5 -yHc 



log.(l+a) t 



and so on, where in is an arbitrary quantity which 



is independent of a; and (2) will become N = l+^^+ * 



log.N\» 1 /log.N 



m ^1.2 



"*" L2i3 V ~wT~ / + &c - ( 4 )- If we change N into N x , 



then since log.N* = rlog.N, (4) becomes N* = l+ar-^*-f 

 log.N\ 3 x* /log.N 



x 



2 



m « 1.2 



m 



~^~ 1.2.3 \ wa~ / +^ c * ( 5 )- If we change a to a', and 



a /2 a' 2 a' 4 



determine a' from the equation a'--o- + -7 r --j-+&c.:=l J (6), 



and then put l+a'=e, (3) becomes log.e=m, and e will be the 

 base of hyperbolic logarithms. By substituting the value of m in 



i v i , a 2 a 3 

 l+a)=log.^a--g-+ 3--&C 



log-N a: 3 / log.N \ a a; 3 f log.N , 

 +ar log.e + 1.2\log.e/ + 1.2.3 IbgT/ + &c ' ( 8 )» 

 which is the exponential theorem. If we substitute the value 



of m in (4), and put N=e, we get e=l + 1 + ^75+ 



-f-&c, (8'), which can also be easily obtained from (8); calcu- 

 lating from (8') the value of e to seven places of decimals, we get 

 e=2-7182818 + for the value of the base of hyperbolic logarithms ■ 

 so that if we have the equation M=e r , by the (ordinary) defini- 

 tion of logarithms sr=log.M when e is taken for base, that is, 

 (since e= the base of hyperbolic logarithms,) z~ the hyperbolic 

 logarithm of M ; we shall denote the hyperbolic logarithm of 

 any quantity by writing L before the quantity, so that I*N, 



