124 Mr. Hellins on the Rectification 
17. But Mr. Landen discovered another and a better me- 
thod of computing the difference in question, which is briefly 
this : 
If from C, the center of 
an hyperbola, CP be drawn 
perpendicular to DP, a tan- 
gent to the curve in D ; and 
if the transverse semi-axis 
CA be = m, the conjugate 
semi-axis = ?z, the perpen- 
dicular CP =.p ; and if/ be ^ 
mm — tin i pp 
put — * y HIlCl % — — • 
r 2 m m 9 
then will the difference between the length of the tangent 
DP and the arch AD be universally expressed by d — the 
fluent of Zy ; ( nn ^”f z _ zz y where d is a constant quantity = the 
fluent generated while % increases from 0 to in. 
If now — be written instead of z, the above fluxion will be- 
come 
PPP 
V( m'n'+2fn~Y agreeing with Mac Laurin’s expression 
of the same thing in Art. 804 and 808 of his Fluxions, where 
the transverse and conjugate semi-axes are denoted by a and 
h, respectively; and, in the latter of those articles, this expres- 
sion is resolved into and its fluent generated 
pp)V(M+PP) b 
while p increases from 0 to a, or m, is exhibted in series of 
which I have spoken in Art. 3 and 15 of this Paper. 
Mr. Landen has no where, that I know of, exhibited the 
fluent of the above fluxion in series, but in the following 
manner.* Denoting the fluent of generated 
* See his 2d Memoir, Art. 5. 
