128 
Mr. Hellins on the Rectification 
preceding Art. over that which was discovered by Mac Laurin 
and Simpson appears; and we see that Landen had good 
grounds for valuing his method, or, to express myself better, 
one of his methods of solving the Problem which I have now 
under consideration, although it cannot be truly said that he 
finished his work. 
21. If indeed the hyperbola were equilateral, then, n being 
m, the fluxional expression —r. — "77! becomes = 
PPP PPP . i P* i 3 P* , 3-5P 11 . 
V(wi 4 — p*) mm * ' s 2m 4 1 2.4 m s > 2 .4-.6m' LZ, ^ " ’ 
and we have 
© 
— x : — 
mrn 3 
3 P* 
3 5 P 1 
2.7m* * 2.411m 8 1 2.4.6.157)2 
n, &C. 
And taking p = m V 
, which, in this case, is 
, the 
1 + V2 
3 + V8 5 
= mV I — we shall have = 
rate of convergency of the geometrical progression which will 
have place in this series ; and this rate of convergency, toge- 
ther with the simplicity of the numeral coefficients, will render 
this series eligible for numerical calculation in preference to 
either of the other series. 
PPP 
., when p 
m 
V 
22. The fluent of ■— — ■ - 
\\mm — pp) v( ww + PP) 
»+V , (mOT +»«) ), ^eing obtained in converging series , whatever 
be the ratio of m to n ; let this particular value of it be denoted 
by <t>, (to distinguish it from the general value denoted by 
and substituted in the general equation d— cp — DP— AD, in 
Art. 17, and we shall have d — <!> = DP — AD ; and this value 
of DP — AD being substituted for it in the particular equation 
L = d = 2.DP — ■ 2AB «f- n — V { mm + tin), in the same 
