H4 
Mr. Hellins on the Rectification 
5. If now we put x = C 3 H we shall have 
uu 
ee ee — uu ■, ee — uu 
xx = , eexx = , and eexx — l = 1 = 
I — UU ’ 
ee—uu — i + mm ee — 1 
1 — uu 
l — UU 
1 — 
uu 
ee 
1 — uu 
uu 
I — __ — 1 -f uu 
ee 
1 — uu 
eexx — x ee — 1 
-■ — - ; we shall also have xx — 1 = 1 
I — MM ’ I — UU 
; and thence, by substitution, 
(I MM 
ee / 
o- 
1 — MM 
I \ 
ee 
uu 
— r — = — , and consequently 
— i)mm mm 1 J 
XX— I I —UU 1 MM 
i ee xx — x \ e 
(«— 0 
JLJL 1 1 C- r w. 
S[——)==t- isJ 
But, since x was put = Vi l _ „ U J> ( see the equation num- 
1 — 
UU 
ee 
bered [V], we have x = 
uu 
ee 
y (x- 
MM \ V ( I — MM) 
ee 
+ 
MM 
a My ( I — — 
ee 
which, by reduction, becomes 
(1- 
1 \ . 
- \uu 
ee 1 
3 
(1 — uu) z 
x v/(x— — ) 
ee 1 
( I — mm) 2 
. And lastly, 
by substituting this value of x in the equation numbered Q£|, 
we have H = x s/ | -y y- p ) = 
(i_ 
e 
(1 -uur x x ~ 
ee ; 
e{i. 
— * the fluxion of the arch of the hyperbola. 
ee I 
6 . Let us now (to simplify the expression) put s = -L and 
* As this result differs from that given by Mr. Woodhoush, in p. 260 of the 
Philosophical Transactions for 1804, I have set down the process at large, that the 
intelligent reader may the more easily perceive where the truth lies. 
