16 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
§ 6. Application to the Equilateral Hyperbola. 
/ d x 
-^3 yj 1 -J- X A j 
which expresses the arc of the equilateral hyperbola, we may put 
1 -j- — V X -j- 1 , 
whence 
x 3 — v 2 x — 2 v = 0, 
we have therefore 2 v — r, and making- this substitution, 
- x — r — 0. 
Also 
v' 1 -]- x A r , I 
X 2 2 x x 2 
S 
V] + X \dx = -J-S — + S— . 
cL x 
x 2 
Now we have 
and 
r „ d x r dr dr 
2 x 2 ' r 2 ’ 
s — — 
3_ 
_ _ 
r 
X 
r 
1 
r ,dx 
d r 
— 
~4 
s 4/1 
+ 
x 2 
*.d 
X = 
d r 
~2 
.dr 3 
+ 4 - 
••• s f 
a/ 1 
+ X 4 
X 2 
. d x 
= 
so that if three abscissae of the equilateral hyperbola are roots of the equation 
r 2 
x 3 — — x — r = 0, 
o 
the sum of the arcs = j r + C, which is the theorem which I originally met with 
concerning the hyperbolic arc*. 
It will be seen how very simply and directly we are conducted to it by the present 
method of investigation. Next let us suppose 
whence 
Put v — — 
y / 1 -j- X* = V X 3 + 1 . 
x 3 — -}- ~~ — 0 . 
V 1 V 
Philosophica Transactions, 1836, Part I. p. 185. 
