of the Hyperbola. 149 
greater than ^3. When a is a large number, about as many 
terms of the single series as of the pair must be computed to 
have the results true to the same number of figures ; yet the 
operation by the pair will be by much the easiest. 
33. If both sides of the equation 
c m, 
See. 
A , B £, D 
a “T a 3 a? T ^ 
a v 
7 . * b . v 
« L -/ x — Ts + 
&c. 
were divided by a, and if “ were put = 'd, and — = b; then, 
(since H. L. of b would be = — /,) we should have 
, d r 1 — Abb + Bb* - Cb 6 + Bb\ &c. 
1 — - l x : cd>b — Cb* -{- yb 6 — « d6 8 , &c. 
where 'd denotes the difference between the lengths of the 
infinite arch and the asymptote, (and the difference also be- 
tween the values of the ascending and descending series for 
computing the arch,) of an hyperbola of which the semi-axes 
are 1 and b, respectively. 
36. It was observed in Art. 12, that the fluxion there given, 
of an hyperbolic arch, is as capable of transformation as that 
which has been commonly used for the rectification of an 
ellipsis : so also are those which I have used in the Philos. 
Trans, for 1802, p. 451 and 455, and from which series con- 
verging by the powers of )*, &c. may easily be obtained ; 
but to treat of such transformations is not only foreign from 
my present design, but would extend this paper to a consi- 
derable length. I shall therefore only point out, by a few 
examples, the great advantage, in many cases, of computing 
by descending series, and then conclude. 
37. Example 1. The transverse and conjugate semi-axes 
mdcccxi. U 
