﻿Exponential and Logarithmic Theorems. 41 



I+2) "(1+3) • '* X ( 1 +M3lj' and by (a) wg shall have M 

 l4-^-(i+^-(l+l)-(l+^-'-(l+^-^)'(l + 



2/ \ i_r 3/ V^M \ i_r 5/ \ i ^2M~2/ \^2M-1 

 &c. (b), by putting M=M-6-f &=(M-&)^l+j^J andM 



fe=(M — fr— c)-(l + M , — ), and so on, (6, c, &c. being arbi- 

 trary numbers,) we may obtain an equation which is more gene- 

 ral than (b); by (9) we may find the hyperbolic logarithm of M, 

 by rapidly converging series, as in the case of the hyperbolic log- 



arithm of 2. (16) is easily changed to L^l+-J=2 



1 



. l+ 



2 i 



r 



+ 



1/ 1 \3 1/ 1 



r s{-^Y + 5{~2q) 5 + &C -\> (19). a «d if we put r=l, (19) 



T T 



n /i 1/ 1 \ 3 1/ 1 



becomes L^+-;=2^^ TI +3^-J +l\gf£i) +&c 

 (20), which can also be applied to find the hyperbolic logarithm 



of M, as given in (6); thus since we have L.2—L. (^l-f^i + 



L. 



2, 9 = 3, we get by (20), 



(l-f-oj, if we put successively q 



2 G + l(^) 3+ K?) 5+&c -) j whose sum = L - 2== °- 693i47 +> 



also if in the formula M=(M-i).(l+jjj^), we put M=10, 

 6=2, we get 10 = 8 (l+j) =2 3 (l+4J, •'• L.10 = 3.L.2 + L. 



1\ /l 1/1\ 3 1/1 



I + 4) =2.079441+2 (5+3(9) +5(9) +&c), by (20), .'. 



L. 10=2.302585+, hence the modulus of the common system 

 of logarithms is easily found, and we shall have (the modulus) 



m=j— ttt= 0.43 4294+; and it may be observed that (20) will gen- 



M 



Vol. xlviii, No. 1 — Oct.-Dec. 1844. 6 





