of the Hyperbola . 1 25 
while p increases from 0 to m, by L, (which I denote by d ,) he 
says that, when the abscissa CB is=fflxv/(i-l- -r , — r], 
J v ' 1 v (mm+nn) I 
(at which time the ordinate BD is = n x %/i-r — 7 , and 
v v \v (mm+nn) I ’ 
! yj {mm-\-nri) , 
the tangent DP = f {mm + ««)), then L, (or d,) is = 2 DP 
— 2 AD -|- n — f {mm -f- nn ) = n + V ( mm "T nn ) — 2 AD. 
18. Now, in order to compare this method with those of 
Mac Laurin and Simpson which have been described above, 
we may proceed thus : ( which > for the sake 
of brevity in a subsequent use of the fluent, I denote by <p,) is 
PPP 
i ny/ (nn-\-pp) 
it is also 
_ PPP 
nyj{mm—pp) 
1 
1 — 
PP 
-L. 
3 
2,4m 4 
3-5 P 6 
+ 
3 5-7/ 
2.4.6m 6 ™ 2.4.6.8m* 
&C. , 
PP _ 
2 nn i z.\n 4 
3i> 4 
3-5P 6 
4 
3 -5- -7P S 
2.4, 6n 6 11 2.4.6. 8/i 8 
, &c. ; 
the one series proceeding by the powers of—, the other by 
the powers of ~ ■; which geometrical progressions, assisted by 
numeral coefficients, it is obvious, will have place also in the 
fluents as they are exhibited here below. Thus, 
The fluents of -77^— -77^—— -r, & c - ^ehig de~ 
\J{nn-\-pp )’ \J (jin-^pp)’ \ (iin-ppp) o 
noted by A, B, C, &c. respectively, we shall have 
PV{nn+pp) -h«h.l. (■£• + ^(i+Sj) 
A 
C 
E 
p 3 V(nn-\-pp) — 3 wm A 
4 5 
pW ( nn +PP) ~ 5 nn B 
“ __ -, 
p 7 \/ (nn J r pp) — 7 nn C 
__ _ - , 
p 9 \/ (nn+pp ) — gnn D 
10 
&C. 
&c. 
