702 PHILOSOPHICAL TRANSACTIONS. [aNNO I796. 



By these expedients the sum of the 3 new series, which is equal to the hyper- 

 bolic log. of 10, may quickly be found. 



p. s. Containing an Improvement of Mr. Emerson s Computation 0/ the Hyperbolic 



Logarithm of 10. 



7. Since the above paper was written, on looking into Emerson's Fluxions, I 

 have found, at p. 137 of the first edition,* another computation of the hyper- 

 bolic logarithm of 10, which is preferable to Mr. Jones's, on account of the 

 swifter convergency of one of the series used in it, as will appear presently. 

 These series also admit of a transformation to others, by which the constant 

 divisors 81 and 64009, used by Mr. Emerson, are exchanged for 40 and 32000, 

 while nearly the same rate of convergency is retained ; which is another re- 

 markable instance of the utility of transformations of this kind. 



Mr. Emerson, considering the number 10 as composed of 

 ^o^ ' = -7b X (^fTr,)^' 3"*^ using the same algebraic series as Mr. Jones used 

 on this occasion, finds the hyperbolic logarithm of JO to be = 10 l. of 



4 ' IC 





1000 



20 . ^0 I 20 I 20 - 



9 ~^ 3.g.81 "'~ 5.9-81' "" 7-9.81^ ' ^" 



4. ii 4. L?_i9__ + __iLLi!'__ + _li-?i_ &c 



~ 233 ' 3.2J3.64-00y ' 5.253 .64.009« ' 7.253.64009'' 



where, instead of a series converging by the powers of -l,-}- as in Mr. Jones's 

 calculation, we have that which converges by the powers of - > ■- or above 4 

 times as swiftly. But what renders this very swiftly converging series still more 

 useful is, that it admits of a transformation, by the theorem in article 4, to an- 

 other series which converges by the powers of ^ , by which the numeral cal- 

 culation is greatly facilitated. 



8. For the two series in the preceding article (the sum of which is = h. l. of 

 10), are evidently = 



"9~ ^ ■• ^ + 3. SI + TTsF" + 7.81^ ' ^^' 

 IS 9 9' 9' 



^"'^ 253" ^ • ' + 3 . t.4009 + 5 . 6+009' ■*" 7-64009" ^^' 



And these two series, when transformed by the theorem above-mentioned, and 



the terms abbreviated, become 



9 A , 2b 3c , 4d - 

 -Z. &c. 



4 3 . 40 5 . 40 7-40 ' 9 . 40 ' 



a •il-'^— _ -^M , 2 ■ 9 b 3 ■ 9c , 4 .9° gjg 



^ 32000 3 . 32000 "■" 5 . 32000 7 . 32000 "*" 9 • 32000' 



Which series admit of some other abbreviations similar to those pointed out in 

 * See also page 197 of 3d edition. — Grig. t Sec article 3. — Grig. 



