Exponential and Logarithmic Theorems. 39 



\-\-a p V — Q P 



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



fj' = r or^=5'+r, (15) becomes log. {q-\-r) = log.g^4-2OTf o i y. 

 +3(2^) +5(2^) +&c.) = w2L.(y4-r), (16), andif we 

 put r = 1, we get log. (^^) =2m (g-^^ + ^^^i) + 



^(2^1+3(2^) +5(2^) +^'=-)' (1^> ^^"^^ ^^^ 

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



weputg'=lin(18), wehaveL.2=2(o4-3(3) +5(3) +&c.j, 

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



in(18), wegetL.3=L.2+2(^+3y ■\-J\^-^ +&c.);andif 

 we put 5' = 4, (since L.4 = 2L.2,) we get L.5 = 2L.24-2 ( g+ 

 3v9/ '^5X91 +*^^v ' ^^^ if 5^=6, (since L.6=L.2+L.3,) we 

 get L.7=L.2+L.3+2(Y3+3(y3) +5(13) +&c.), 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 2i=l-\-l = {l-\-y).{l-{-z) = \-\-y-\-z-\-yz, 

 1-z 

 we get y=TT — ' •'" ^^ assume z=^, we get y=J, and 2= 



^l+^)-(^l+3). In a similar way if we assume l+J=(l+y). 



i-z 

 (l+z) we get y=YZ^^ ^"^ ^^ ^® assume z=i, y=h so that 



^~\^'^3/ V"^!/ V"^5/' ^^^ ^^ ^® change^ into ^ in the 



*-^ . 1 1 



formula y^-^j^- and assume 2;=^, we get y=«;, so that 2= 



