QUADRATURE of the CONIC SECTIONS, &c. 315 
of thefe inquiries, it appears, that the terms of the feries, un- 
der its firft form, (Art. 43) are exactly the {quares of the cor- 
refponding terms of the former feries, under its firft form 
(Art. 36.), fo that the one being written thus, 
==P— (Tey Tee) ++ ee tT (m+ T (mes) + Tinta) ts &es) 
Ss 
the other will be 
SSP (Ta) AT) «FT + Ting 1) FT na) +, &-), 
and here P and P’ are put for the parts of the two expreffions 
which do not follow the law of the remaining terms, but T(1), 
T(2), &c. denote the fame quantities in both. Now, as each 
term in the former feries has been proved to be greater than 
one-fourth of the term immediately before it (Art. 39 ), each 
term of the latter muft be greater than one-fixteenth of the 
term immediately before it ; and this is one limit to the rate of 
convergency. 
_ AGarn, as it has been proved (Art. 40.), that in the 
, therefore, fquaring, we have 
ool : T’(n+1) 
firft feries Ton+2) fe bie a 
4 
T’(n+2) ot. Now this quantity is a third proportional 
to Tn) and T*(n41)3; hence it follows, that the greater limit of 
the rate of convergency in-the two feries is the very fame; 
_ that is, each term is lefs than a third proportional to the two 
terms immediately before it. 
As thefe limits to the rate of convergency differ from 
thofe of our fecond feries for an arch of a circle (Art. Pos), 
only by the leffer limit in the one cafe correfponding to the 
“greater in the other, and the contrary, it is fufficiently evident, 
Vou. VI.—P. II. sr that 
