Exponential and Logarithmic Theorems. 41 



l+gj'll+o) • • • X (l-+I|ir]r)j and by (a) we shall have M= 

 1\ / 1\ / 1\ / 1\ / 1 \ / 1 



&c. (6), by puttingM=M-& + 6 = (M-6)-(l+jj^^j andM- 



c 



6=(M — 5— c)-( l4- iy,_ , —\, and so on, (&, c, &c. being arbi- 

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

 ral than (6) ; 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(1+~)=2 



1/ 1 \3 1/ 1 \5 



2o 



1+— 

 r 



(19), and if we put r=:l, (19) 



3\ 2^/ -5^ 2^ 

 r T 



becomesL{l+i)=2(2^^+^(.^-J%^(2-^^^ 



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



of M, as given in {h)\ thus since we have L.2=L. ( l+o) + 

 L. ( l+o)> if we put successively ^'=2, ^=3, we get by (20), 

 L-(l+^)=2Q + gQ)V|Q)V&c.), and l(i+|) = 



^(7 "'"3(7) +5(7) +<^'^-)' whose sum =L.2 = 0.693147+, 

 also if in the formula M=(M-6).f l+^^^J, we put M=10, 



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



(1+2) =2.079441+2 (J+g(^)V^Q)V&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) 



1 

 m=j—j^ =0.434294+; and it may be observed that (20) will gen- 

 erally be found more convenient in practice when applied to M, 



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



