3 io NEW SERIES for the 



of a circle, (fee Art. 8. and 9.), only the greater limit in the 

 one cafe correfponds to the lefTer limit in the other, and vice 

 versa, 



41. We might now, from thefe limits to the rate of conver- 

 gency, determine two limits to the fum of all the terms of the 

 feries following any given term, by the mode of inveftigation 

 employed at Art. 10. and Art. 11. in the cafe of the circle; 

 but the refult in both cafes would be found to be the fame, with 

 the difference of the fign < for > , and > for < ; that is, we 

 would find the fum of all the terms following any term of the 

 feries, to be greater than one-third of that term, but lefs than 

 a third proportional to the difference between the two terms 

 immediately before it and the latter of the two. 



42. Upon the whole, then, our firft formula, for the quadra- 

 ture of an hyperbolic lector, may be exprefled as follows. 



Let s denote the area of the fector, and put p for -. — — —. 



r r ab s — ord s 



Then, 



1 



1 2 abs s 



2 4 Q 16 



P —I p — I P —1 p — I 



s ~~ ord s \ L ^ 1 L T 4 1 T 8 1 



iy + i / + i p* + i p i6 +i 



+ T W -fT(,„ + .) + Ri 



where T( m ) and T( m +i) denote any two fucceeding terms of the 

 feries, and R the fum of all the following terms *. And 



here 



* The fame feries may alfo be put under another form, which it maj not be 

 improper to notice briefly, on account of the facility with which the terms may 



be 



