Mr. Hellins on the Rectification 
126 
and hence, 
i 9 * <P = “ x ; A-f 
2 mm 
B + — ,C + 
■ 7 .. Amy 
3-5 jp . 3-5-7 
2.4 m* 1 2.4.6m 6 ' 2.4.6.8m® 
PPP PP 1 Pf 
E, See. 
And the fluents of - ■ FFF ,, - v rr, -77- ;/ r rr, &c. being 
<J(mm—ppy^(mm—ppy‘>J(mm — ppy o 
denoted by A', B', C, &c. respectively, we shall have 
A' — 
mm x cir. arch of which rad. is i and sine — , — p^/[mm— pp) 
Tj, 3mmA' — p J \/ (mm— pp) 
“ - » 
5mmB' — p s \/ (mm —pp) 
— g- — > 
c # 
D' 
ymmC'—p 7 \/(mm—pp) 
c- 
3-5 
D' 
3-5-7 
E', &c. 
gmmD'—p 9 if (mm — />£) 
&C. &C. 
and hence, 
O __ . A / 8 TV I 3_ 
* ^ « * 2«« 2.4W 4 ^ 2.4.6« 6 ■*"' f 2.4.6.8?i 35 
Each of these series begins and increases with p ; so that 
neither of them needs any correction. 
19. Two series being thus obtained as expressions of the 
general value of the fluent of (denoted by 
<p,) the next thing to be done is, to ascertain the rate of conver- 
gency of each when the abscissa CB is *= m </(i 4 * 7 — - — 
o J v v i ( mm nn) l 
Now to this value of the abscissa CB, the corresponding value 
of the perpendicular CP =^> (as appears from the equations in 
Art. 17.,) is = tnV{- 
pp n 
4V(mm4»t!) 
— , and — = 
7 nn 
j. We therefore have, in this 
, which are 
mm 
case 
5 mm ” n+\/(mm-{-nny “““ nn nn-\ny/ (mm-fw«)’ 
the respective rates of convergency of the geometrical pro- 
