127 
of the Hyperbola, 
gressions which (assisted by numeral coefficients) will have 
place in the first and second series given in the preceding 
article. 
20. It is now 
easy to compare these rates of conver- 
gency with that which has place in the series in Art. 15, thus : 
Putting 1 instead of m, and b instead of n, 
the rates are these. 
viz. 
In 1st Series. 
b 
In 2d Series. 
1 
In Art. 15. 
1 
6+v'(I+^) , 
bb-{-b\/ (i + bby 
i+bb * 
And writing 1, 2, 
1 
3, &c. successively instead of b , we have 
1 1 
1+V2 ’ 
2 
1+V2 3 
1 
2 * 
1 
2+ Vs ’ 
3 
4+2V5 * 
1 
5 s 
1 
3 + -v/i°’ 
9+3V lo * 
IQ 5 
Sec. 
&c. 
See, 
In all which cases it is evident, that the calculation by the 
series given in Art. 15 will, on a double account of simplicity, 
be much easier than by the second series in the preceding 
Art. notwithstanding its greater rate of convergency. 
Let us now write &c. successively instead of b , 
and we shall have the following rates 
In ist Series. 
In 2d Series. 
In Art. 15. 
1 
i+Vs ’ 
4 
i+Vs ’ 
4 
S’ 
1 
i-f Vio* 
9 
iV + IO* 
9_ 
io* 
1 
16 
1 6 
i+Vw’ 
i-j-v/17* 
&c. 
&C. 
See, 
Here the great advantage of the first 
series given in the 
