129 
of the Hyperbola. 
Article (in which equation DP == \/ we have d — 
2 d — 2$ -f- n — s/ ( mm -f- nn), and hence L = d = 2® 4- 
✓( * + — n , which is the difference, or quantity 
sought. 
23. I come now to make a comparison of Mr. Landen’s 
last mentioned method of computing the difference in question 
with some of my own methods. 
We have already seen in Art. 20, that, when the conjugate 
axis of the hyperbola is greater than the transverse, Landen’s 
method is not wanted, since the operation by the old series 
will, in general, be easier. The comparison therefore, now 
to be made, is only in cases when the conjugate axis is equal 
to, or less than, the transverse axis. 
It appears from Art. 9,11, and 17 of this Paper, that, when 
m is put = 1 , and x is taken = m \/ ( 1 -4 - t 7—- 7 1 , the value 
of ee uu is — I — the powers of which fraction form 
the geometrical progression which will have place in the series 
denoted by G, from which H, or the arch AD is quickly ob- 
tained. And, with these values of m and x, we have, by Art. 
in, — — — - the powers of which fraction form the 
geometrical progression which will have place in the series 
denoted by <t>. When n is taken = 1, the former of these 
algebraic fractions is = — 1 ^ /2 , the latter is = — As the 
one fraction increases with w, and the other decreases, it is 
easy to find that value of n which shall make them equal, viz. 
?z = -§ v /( — 2 -{- \l 20 ) = 0786, &c. ; and, with this value of 
n, each of these fractions is = — — = 0-381966, &c. Thus 
it appears, that, n having any value greater than ~ \/( — 2 + 
S 
MDCCCXI. 
