308 NEW SERIES for the 



an equation which exprefles the property we propofed to inves- 

 tigate. ** 



38. We have now only to fuppofe n in this formula to have 

 thefe values, 2, 4, 8, &c. fucceflively, and to fubftitute inftead 

 of the terms of our feries 



1 2 A _ 1 , i . 1 . 1 iio\ 



- zz — [ tan - s + _ tan - s + - tan - s +, &c. 1, 



s tan s \ 2 2 4 4 8 /' 



their values as given by the formula, putting alfo or , inftead 



ab s 



of tan j, and the feries becomes 



1 1 1 



1 2ab.f (V — 1 ,l/> 4 — i.ip — l_i_c 1 



7 = 1^7- y-i — +2 - — + 4 1 — + > &c l 



V 2 ■ 4 . '8 , 



p + x ^ +' 1 A + » 

 and this is the new form under which we propofed to exhibit 

 it. 



30. Let us now inquire what are the limits of the rate of 

 convergency of this feries ; and in doing this, it will be mofl 

 convenient to refer to the firft of its two forms. Now, from 



the formula — ~ - - - 1 , c — - tan I S, we get 

 tan S 2 tan \ S 2 2 



tan - S = - tan S (1 -f tan* - S). But 1 + tan 2 1 S > 1, and 

 22 2 2 



therefore - tan S (1 + tan 1 - S) > -tan S, hence it follows, that 



2 22 



tan - S > - tan S. Thus it appears, that each term of the fe- 

 2 2 



ries 



