40 Exponential and Logarithmic Theorems. 



(1+4) V+5J V+6/ V"'"7/' and by proceeding in this way, 



we get 2=(l4).(l+^)-(l+^)-(l+^)-(l+^) = (l+ro)* 



18/ V^\9I~V^'^QI V^2VI^ ^\"^38y \"^39y 

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



/ 1\ 1 1/lV l/l\' 1 



L.(l+3J,wegetby(9)L.2=^-^(2/ +3(2/ "^C'+s" 



(5/ / + 3\(3) +(4) +(5) ) ~^^' ^"^ ^° °"' by which 

 means the hyperbolic logarithm can be found by rapidly con- 

 verging series. Also since 3=2 + l=2( l+n] = ( l+oj'll+o) 



•fi+^)=(i+5y-(i4)=(i+i)'-(i+^)'-(i+s)'(i4), 



'2)~V^2l V^3J~V^AI V^5I V^6J V^7 

 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 '^ + ~=\^+2n)'\^'^ 2n -^1 / ~ 



^ + ;i^]-(l+z:;^)-{l+4;^)-(l+4;^ 



An/ \ An- 

 if M denotes any positive integer greater than unity, (since M= 



M~l + 1,) we get M=(M-l).(l+jj^^j,andifM-l is great- 

 er than unity, we have M-l=(M-2).f l+j^^5],andif M-2 

 is greater than unity, we have M — 2=(M- 3).( l+:jrv3o)j and 

 so on ; and by these reductions we shall finally get M= f I+t) • 



