QUADRATURE of the CONIC SECTIONS, &c. 275 

 fince fee \ A < fee A, and confequently 



1 -I- feci A i+iecA' 



therefore C -^44 > ^A and tan 7 A > t "^ Li # X tan i A. 

 tan i A tan A 4 ^ tan A 2 



In this expreflion, let \ a, ~ a, \ a, &c. be fubftituted for A, 

 and let the refults be divided by 8, 16, 33, &c. ', then we get 



• „ 1 "7 tan -7 <? v. 1 ^ 1 



. can "7 ** 



4 4 



&c. 



I I , I . I 



from which it appears, that in the feries, - r: f- z. tan - # 



« tan a 2 2 



+ i tan - a + I tan I a + \ tan \ a -f- &c. 

 44 8 8 10 10 ' 



each term after the third (that is, after ~ tan \ a), is greater 

 than a third proportional to the two terms immediately before 

 it, taken in their order; and this is another limit to the rate of 

 convergency of the feries. 



10. The limits which we have found to the rate of conver- 

 gency of the feries, enable us alfo to affign limits to the fum 

 of all the terms after any given term. Let the feries be put 

 under this form,. 



s = dr a + -2 tan I a + l tan I a - • • + T w + «*■+* 



Vol. VI.— P. II. Mm where 



