﻿Exponential and Logarithmic Theorems. 39 



1+a p p-q P 



put j3^=-, wegeta==— j— , and (14) becomes log. - = 2 m 



(j+,+l$+J '+8^J) 3 +&c.) -*U| (15), and if p 



q = r orp=q+r, (15) becomes log. (q+r) = log.g-+2m(n— — 



1/ r \ 3 1 



+ 3\27+W +5(2^) +ifec.) = mL.(y+r), (16), and if we 



<7+!\ « / 1 1/1 



put r = 1, we get log. [~ J = 2m ^ + g^j + 



5(^+1) 5 +&c) = mL.(^),(17), .".L.fo+l) 



9 



1 1/ 1 \ 3 1/ 1 



+ 



2 \2qT'L + 3\2qTll +512^1) +&C '/' (18)> ShlCe the 



logarithm of unity is equal to zero in any system of logarithms, if 



we put?=lin (18), we have 1.2=2(3+3(3) +g[^Y +&,c. 

 which gives the hyperbolic logarithm of 2; then if we put q = 2 



in (18), we get L.3=L.2+ 2(5+3(5) ' + l\l) +&c.) ; and if 



we put q = 4, (since L.4 = 2L.2,) we get L.5 = 2L.2+2(g + 



1/1\ 3 1/1 



3\9 



+ 5\9/ + &c ) > and if ?= 6 > ( since L.6=L.2+L.3,) we 



1 1/1\ 3 1/1 



getL.7=L.2+L.3+2^ I 3+^ T 3J + 5 ^j +&c.J, and so 



on, to any extent ; if we add the hyperbolic logarithms of 2 and 

 5, we get the hyperbolic logarithm of 10, and if we divide unity 

 by the hyperbolic logarithm of 10, we get the modulus of the 

 common system of logarithms. 



Again, if we assume 2 = l + l=(l+y).(l+^) = l+y+s+y^, 



\-z 



we get y= j-T— , .'. if we assume z — i, we get y=% 



+2J'\1+3J. In a similar way if we assume l+£=(l+y). 



i-z 

 (l+5r) we get y=^-—, and if we assume *•*£» y=±, so that 



2= \ 1+ 3/ \ l+ z)\ l +5l> and if we chan S e i int0 i ^ the 



r , i-« , 1 1 



iormula y=s ^ . and assume z=q, we get y=~ so that 2 



