﻿40 Exponential and Logarithmic Theorems. 



1+4 W l+5/(l+6/ 1 1 "^"7/ J an( * by proceeding in this way, 

 we get 2=( 1+ i).(l + g).(l+i)-(l+g)-(l+g) = (l+i 



1 +s)'( 1 +^)=( 1 +^)-( 1 +s) x --' x ( 1 +^)-( 1 +^l> 



and so on, to any extent; hence since L.2 = L.(l+o/ + 



/ 1\ 1 1/1\ 2 1/1\ 3 1 



L.^l+3J,wegetby (9)L.2=2~2^j +3^/ -&C.+3 



i\3\ i//i\4 m\4\ 1 1 1 i//i\« (iy 



3/ /-4W2/ +I3) J+ &c -=3+i+5-2U3J + U) + 



+ o((-i) +b) +(i) J -&c. and so on, by which 



5/ / T 3V\3/ ^ 14/ ' \5 

 means the hyperbolic logarithm can be found by rapidly con- 



1\ ( 1\ / 1 



verging series. Also since 3=2+1=2^1+^ ^l+gj-l 1+3 



and so on, we may find the hyperbolic logarithm of 3, in a sim- 

 ilar way by a rapidly converging series, without making it de- 

 pend on the logarithm of 2, as is commonly done by using (18). 

 From what has been done, it is evident that if n denotes 



any quantity, we shall have 1 + -= (l+^J \ i +2n +1 



1 +5 ! )-( 1 +stt)-( 1 +sW)-( 1 +4ST3-)= &c - w; and 



if M denotes any positive integer greater than unity^ (since M 

 M - 1 + 1,) we get M=(M- 1)^ 1 + jj^j J , md if M- 1 is great- 

 er than unity, we have M-1 = (M— 2). I 1+^—5 Land if M-2 

 is greater than unity, we have M— 2=(BI- 3).^l+jjj3gl, and 





so on ; and by these reductions we shall finally get M 



1 

 1 



