308 NEW SERIES for the 
an equation which exprefles the property we propofed to invef- 
tigate. > 
38. WE have now only to fuppofe z in this formula to have 
thefe values, 2, 4, 8, &c. epetieey and ‘to fubftitute inftead 
of the terms of our feries 
I 2 
a = Se es =+t 5 5 ma 
; (un ies an 4 Atl an Deel ‘fre 
/ tan's 
their values as given by the Goal) putting alfo 2°°* inftead 
ae 
of tans, and the feries becomes 
Ste 
_ sab?“ hblty thems 4 tah 5g, 
—ord's t= 2 - 4 TFS 
ee ee 
and this is the new form under which we propofed to exhibit 
it. . 
39. LET us now inquire what are the limits of the rate of 
convergency of this feries.; and in doing this, it will be moft 
convenient to refer to the firft of its two forms. Now, from 
: tT’ I I ope Be 
‘formula ee le = tan > S,* we get 
the «(formule nS 2tan=S 2 Bigs g 
pT yk 
tan 2 S= Ltan$ (1+ tan” = 8). But 1 + tan*® — S> 1, and, 
2 2 0) 
therefore = lian S (1+ tan’ - 7) > ~tan S, hence it follows, that 
tan £S>2tan$. Thus it appears, that each term of the fe- 
2 2 
ries 
