of the Hyperbolic Logarithm of 10. 137 
1 + p7 + -pT" + -pi r» &c. We therefore now have 
3L. 2 -{- L. — equal to the sum of these three series, 
2 x : 1 + + 7JJF’ &c * 
7 x : 7 + 7^7 + WIT - + &c. 
9 x : 1 + 3.81 
— * - ~ r H ~-r, &c. 
5.8i a ■ 7.81 3 9 
which sum is also equal to the hyperbolic logarithm of 10. 
4. The form to which Mr. Jones’s series are now brought 
is evidently the same with the general form a x : 4* m ' +~ i 
+ 
+ 
&c. the value of which, while m and n 
m + zn • m $n 
are affirmative numbers, and x sufficiently small, will be 
given by the series a x : 
m. m + n. L — x n \ 
+ 
n. zn. x ln 
n. zn. 3 n. x 3 
m.m + n. m + zn. i — x"\ 
&C. 
1. m + n. m -j- zn. m + 3 n. 
And this series, if we call the first, second, third, &c. terms of it 
A, B, C, &c. respectively, and put - x _ — = ■%, will be more 
concisely expressed thus ; a x : --_L= nzA ' znzB 
m -J- n 
m -f zn 
H ^-7—7—, &c. which form is well adapted to arith- 
metical calculation. 
Now, by comparing the three series at the end of the last 
article with the general series here given, we shall find that, 
in the first and last of these series, the value of m is 1, and in 
the second of them it is 3. The value of n in the first and 
* See Phil. Trans, for 1794, Part zd. p. 218, where this matter is more fully ex- 
plained. 
MDCCXCVI. T 
