Exponential and Logarithmic Theorems. 37 



fls \ y^ I 11 \ y^ I 3a* \ 



y-&c. j + j;2\«' -a^+j2a*-&c.j +1:2:3 ^«^ " T" + ^""'1 



y' 

 + j-Q-0-7 (a* — &c.)+&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, 



f a^ a^ a* \ 



hence we must have log.(l+a)— mla— -2- + -o" — "T+&c. 1, (3), 



11 /log.(l+a)\=' 3a* 



and a2 _cj3 _}_ _^4 _ ^c. = ^ — j , a^ - -g- + &c. = 



/log.(l+a)\3 



I 1 J and so on, where m is an arbitrary quantity which 



log.N 1 

 is independent of a ; and (2) will become N=l-J- '^Tq, 



/log.N\ 2 1 /log.N\ 3 



[ml +12731 mj +^^- W If we change N into N^ 



log.N x^ 

 then since log.N"' = arlog.Nj (4) becomes N'' = l+:r +r2 



/log.N\2 2;3 /log.N\ 

 \1n~j '^h2.3\~m~l 



+&C. (5). If we change a to a', and 



determme a' from the equation a' — -o" + TT — "7" +^'^' ~ ■''^j (^)> 



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 a^ a^ \ 



(3)and(5), they will become log.(l+a)=log.e{ a- -o-+-o"-&c.)) 



, _ log.N a;2 /log.N\2 x^ /log.N\ » 



which is the exponential theorem. If we substitute the value 



of m in (4), and put N=e, we get e^l + l + j-^ + j-^^+ i 234 



+&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—2-7 1828 18 + for the value of the base of hyperbolic logarithms ; 

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

 tion of logarithms 2;=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 L.N, 



