VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. 701 



2 .81 _ 4^ 1 8b _ 12c 16d „ 



80 5 . 80 "^ 9 . 80 13 . SO '' 17 . SO' ^" 



. 2.81 4a ■ SB 12c i6d „ 



^" 9.3.80 7 . 80 "*" 11.80 15 .80 ' 19 . 80' ^' 



,2.81 2a , 4b 6c , 8d o 



and — , &c. 



9 . 80 3 . 80 ' 5 . SO 7 . 80 ' 9 . 80 ' 



which 3 series are equal in value to those in art. 3, and to the ftyperbolic loga- 

 rithm of 10. 



5. With respect to the convergency of these new series, it is evidently some- 

 what swifter than by the powers of 80. For even in the first series, which has 

 the slowest convergency of the 3, the co-efficients -*-, ■§-, -i-|-, &c. are each of them 

 less than 1. 



6. But another advantage of these new series is, that their numerators and 

 denominators may be reduced to simpler terms, in consequence of which the 

 arithmetical operation by them is further facilitated. In the 1st and 2d series, 

 every term after the first is divisible by 4 ; and every term in the 3d series admits 

 of a similar reduction by the number 2. The 3 series then, when these reduc- 

 tions are made, and their first terms are also abbreviated, will stand as below 

 (each still converging somewhat faster than by the powers of 80) ; and we shall 

 have the hyperbolic logarithm of 10. 



f 81 A__ 2n _ 3c 4d „ 



I 40 5.20 '" 9 . 20 ~ 13.20 ~T" 17.20' ^" 



1 



^ ^_ J_ _£^ _ 3C 4» c,„ 



40 7.20 ^ 11.20 15 . 20 ' 11) . 2u' 



J. t L ^° _ 30 4.D 



L 40 3 . 40 ' 5 . 40 7 . 40 ' 9 . 40 ' 



The arithmetical operation by the new series is undoubtedly easier than by the 

 original series; yet it is evident, by inspection, that half the number of divisions 

 by 20, in the 1st and 2d series, may be exchanged for divisions by 10, which 

 are no more than so many removals of the decimal point ; and that, in the 3d 

 series, half the number of divisions by 40, the first excepted, may be exchanged 

 fot easier ones, i of them for divisions by 20, and the other 4th for divisions by 

 10. The new series then, still converging somewhat quicker than by the powers 

 of 80, may stand thus : 



81 A , B 3c , 2d . 



_J_ ^_ _1_ oCC 



T^ 11 in I) on T^ 17 in' 



40 5 . 20 ~ 9 . 10 13 . 20 ~ 17.10' 



J 3 A B 3c 2d „ 



^"^ ~^ ~ TT20" + TTTTT) ~ TT^ + IgTw' ^^• 



J 9 A , B 3C , D 5 



And even yet one might still facilitate the computation of the value of some of 



..U.. rr^U 3- 1 + A3. 1— ii. 1 j3. 



the terms. Thus, — is = -^ ; __ ,s = -j^; - is = -; and -^^^ ,s = 



tL, &C. 



