QUADRATURE of the CONIC SECTIONS, &c. 333 



x 



+ ± 



(x l) 2 12 



log*# 



and thus we have obtained a fecond feries for the logarithm of 

 a number, which, by putting t, t', &c. inftead of the fractions 



x* — 1 x 4 — 1 



-, &c. and remarking that the relation which 



x 2 -\- 1 x* + 1 



the quantities t, f, &.c. have to one another is identical with 

 that of the quantities tan/, tan 4- J", &c. (Art. 35'), it will ap- 

 pear to be the fame as our fecond feries for the area of an hy- 

 perbolic fe&or, (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 mull now transform our feries upon principles pure- 

 ly analytical, fo as to fuit it to calculation. And, in the firft 



(X I a X 11 2tf-j-I l(ff-fi) I 

 J — — 

 x 4- 1' x 



-\- 2 X -{- I 



Tt 2 



x(*+y+i* 



if 



we 



