138 Mr. Hellins’s Improvement in the Computation 
second series is 4, and in the third it is 2. The values of a 
are obviously 2 
third. But in each of them 
in the first series, and J- in the second and 
is 
80" 
and 
and 
2.81 
4 a 
8B 
12C 
+ 
16D 
80 
5.80 ' 
9.80 
13.80 
17.80 ’ 
2.81 
1 
8B 
12C 
+ 
16D 
9.3.80 
7.80 • 
1 1.80 
15.80 
19.80 ’ 
2.81 
— — + 
3.80 ' 
4 B 
6C 
+ 
8D 
9.80 
5.80 
7.80 
9.80 ’ 
These values of the letters being written for them in the se- 
cond general form, we have three new series, viz. 
qr ,~n , 
&C. 
&C. 
&c. 
which three series are equal in value to those in art. 3, and 
to the hyperbolic logarithm of 10. 
5. With respect to the convergency of these new series, it 
is evidently somewhat swifter than by the powers of 80. For 
even in the first series, which has the slowest convergency of 
the three, the coefficients j., -Li, &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 first and second series, 
every term after the first is divisible by 4 ; and every term 
in the third series admits of a similar reduction by the num- 
ber 2. The three series then, when these reductions 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 
