310 NEW SERIES for the 
of a circle, (fee Art. 8. and g.), only the greater limit in the 
one cafe correfponds to the leffer limit in the other, and vice 
VETSAs 
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. rr. 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 feétor, may be exprefled as follows. 
abs -+ ords 
Let s denote the area of the fector, and put f for in age 
Then, 
| pea owihe sat adios, Aa 0 ipa pee - 
2abs s r I I 
ord s : 21% ch ie Re Bi 
oe Pk Xp pa oP rg tg eg 
Sr T(m) a5 T(m41) + Ri 
where T(m) and T(m4.1) denote any two fucceeding terms of the 
feries, and R the fum of all the following terms*. And 
here 
* Tur fame feries may alfo be put under another form, which it may not be 
improper to notice briefly, on account of the facility with which the terms may 
be 
