14 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
u — 0 , 
which agrees with the given form by putting r = — u. 
We have now to find the sum of the arcs. 
The arc subtending the abscissa (x = &>) may be considered as equal to it. 
The arc subtending the abscissa y differs from the elliptic quadrant by an arc 
which may be considered equal to the ordinate which corresponds to y. And the 
same with respect to z. 
Let y be the ordinate corresponding to y. The equation of the curve gives 
y = b s / 1 -y 2 ; 
but since 
r 
•Vi 
y 
2 — 
b w 
o 5 
and the arc subtending?/ = Q — 
The arc z has the same value. Therefore 
arc y -f arc z = 2 Q — b 2 a ; 
adding arc x — we have 
Sum of arcs = 2 Q -j- e 2 a, 
(or, since u = — r) 
— 2 Q — e 2 r, 
in accordance with the theorem. 
Ex. 4. Let 1 — e 2 = 
And also let r = 9 — 3 \/ ] 0 = — (MSGSSSl ; 
the roots of the equation 
x 3 + r x 2 + — 1 ^ x — r = 0, 
are 
x — 0*5 = sin 30° 
y — 0-98019 = sin 78° 34' 
z = — 0-99336 = sin 83° 24' 
Entering Legendre’s Table IX. with modulus e — \f ^ r — sin 54° 44' and these 
amplitudes, we find 
arc x — 0"5081 
arc y = 1‘1 446 
arc ^ = 1*1944 
Sum = 2-8471 
