QUADRATURE of the CONIC SECTIONS, &e. 329 
and this is the firft feries which I propofe to inveftigate for the 
calculation of logarithms. 
59. Tue feries juft now found agreeing exactly in its form 
with our firft feries for an hyperbolic fector, (Art. 40.), as it 
ought to do, will of courfe have the fame limits to the rate of 
its convergency, and to the fum of all its terms, following any 
propofed term. As the latter of thefe have been deduced from 
the former, in the cafe of the hyperbola, by a procefs purely 
analytical, and the fame as we have followed in treating of the 
rectification of the circle, it is not neceflary to repeat their in- 
veftigation in this place. The limits to the rate of convergen- 
cy, however, having been made to depend partly upon the na- 
ture of the curve, it may be proper, in the prefent inquiry, to 
deduce them entirely from the analytical formula which has 
been made the bafis of the inveftigation. 
Ler any three fucceflive terms of the feries of quantities 
ei —1 wt —1 
x? +1 xt +1 
evident from the formula, (Art. 58.), that the relation of 
thefe quantities to one another will be exprefled by the equa- 
tions 
» &c. be denoted by ¢, z#’ and?’; then it is 
? 
2218 Y pp .2 2 Jr é 
= t Sp emes (AS 
t t + ? t er 
From the firft of thefe we get 2/ =¢(1-+ 7"), now each of 
the quantities 7, t’, &c. being evidently lefs than unity, it fol- 
lows, that 1-+#2 <2, but > 1, and therefore that 27’ < 21, 
, I : 
and 2’ <?¢; alfothat 27> ¢, and t’> 5 ft Hence it appears, 
in 
