MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
11 
The three variables x y z are roots of 
[I.] <r 3 - a x 2 + (y - a b =p ^-) x + ^ = 0, 
the upper sign applying to the parabola ; a and b being two arbitrary quantities. 
To exemplify this theorem in the circle. — Since 
dx \/ 1 — x 2 = x \/ 1 — x 2 -\- arc sin x, 
the theorem gives 
— 2 r = S .r \/l — -z 2 + S arc sin x. 
The latter term, being the sum of three circular arcs, cannot form any part of the 
quantity — 2 r : therefore we must have this other equation, 
S arc sin x = 0, 
which we propose to verify. 
Ex. 1. Suppose a — y, b = " the equation [1.] becomes 
3 6 9 7 . 12 A 
X' 3 ~ x 2 — — x = 0 , 
25 
and its roots are 
x = 
V = 1 
3_ 
5 
.'. arc sin x = 53° 8 ' = 6 
arc sin y = 90° = & 
arc sin 2 = — 143° 8 ' = 6" 
:. 6 + & + §' = 0 , 
in accordance with the theorem. 
We may assume two of the arcs arbitrarily, and thence determine the third, so as 
to satisfy the theorem. 
4 IQ 
Ex. 2 . Thus, let x — — } y = — , we find a — gy b 
12 
T‘ 
Here it happens that 
a b 48 
X V = ~ 2 ~ = 65 : therefore, dividing the equation 
by the equation 
a 3 b 
xy% — -- y 
a b 
xy = 
Q. ’ 
c 2 
we find the third root 
z = — a. 
