QUADRATURE of the CONIC SECTIONS, &c. 333 
(ish I 
hoa. FS 
- if a 
2 4 2 8 mi 
4 te Cai I ea 0 gall a 
oe Wipe apiosad I are I 44 I ie Z 
g * 4\ > 4 = =, z 
x +I x +i Ee ear! 
and thus we have obtained a fecond feries for the logarithm of 
a number, which, by putting ¢, 7’, &c. inftead of the fractions 
ue Ee 
x7——IT xt—Ff 
; , &c. and remarking that the relation which 
x? aE xt +1 
the quantities 7, z’, &c. have to one another is identical with 
that of the quantities tans, tants, &c. (Art. 35.), it will ap- 
pear to be the fame as our fecond feries for the area of an hy- 
perbolic fector, (Art. 43.). Of courfe it will have the fame li- 
mits to the rate of its convergency, and to the fum of all its 
terms following any given term. Now thefe have been found 
without any reference to the geometrical properties of the 
curve, therefore it is not neceflary to repeat their inveftiga- 
tion. 
62. We muft now transform our feries upon principles pure- 
ly analytical, fo as to fuit it to calculation. And, in the firft 
W—To2 wx? —ax+y 1(w+1)—1 
place, becaufe ( ) = : 4 i 
x +1 
— 
———— = —___—_ if 
a -anpax 9 +2) +2? 
pa we 
