QUADRATURE of the CONIC SECTIONS, &c. 309 
I 
; s, &c. is greater than half the 
’ f 
Cs ele nh 
ries of quantities PRT 5, tan 
term before it; and as thefe, multiplied by the fractions *, > 
&c. refpectively conftitute the terms of the feries, each term of 
the feries, under either of its:forms, is greater than one-fourth 
ofthe'term before it. 
*4o. Acatn, from the formula tan Ss =< tans (a+ tan’ S) 
tan+S I Sige 697 + 
we find Pesca Ri = 1+tan* = S, and fimilarly, pens eis 
tans Is12 tanisS 
I+ tan’ ‘i S. But from the nature of the hyperbola 
tanis ub 
r+ tan? 2S <1 tan’ 2$;-therefore 24029 _2tan 78 ang 
4 2 tan;S tan 5 
tan*+S 
hence tan 2S < 2, 
4 tan S 
Therefore, putting = s inftead of S, 
and multiplying by at we have 
r b pea 0 erry es 
2n n . 
—tan-— sS< AO ot <= gan Bee, 
82 I I 2n 
- tan—s 
n n 
from. which it appears, that each term of the feries, following 
the fecond, is lefs than a third proportional to the two terms 
immediately before it. So that, upon the whole, it appears, that 
the limits of the rate of convergency of our firft feries for an 
hyperbolic fector, are the fame as thofe of our firft for an arch 
Qq2 of 
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