MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
13 
Or, if we suppose the radical to have a negative sign. 
S /‘»*v/ 1 CT = c 
whence the following theorem : If three abscissae of an ellipse are roots of the equation 
the sum of the arcs = 2 Q — e 2 r, Q being the quadrant of the ellipse. 
Ex. J. Let e — 0, or the ellipse be a circle ; the theorem then assumes this form : 
If three abscisses of a circle are roots of the equation 
the sum of the arcs is a semicircle ; the truth of which has been demonstrated pre- 
viously (vide page 4.). 
Fagnani’s theorem becomes illusory when e = 0: it is therefore interesting to ob- 
serve that the present theorem, on the contrary, has a real application to the circle. 
Ex. 2. Let e have any value, and r = 0 ; then the roots are 
which is therefore the value of the constant. 
Ex. 3. When x is not actually = 0, as in the last example, but has an indefinitely 
small value = u, it will be found that the values of y and z differ from 1 and — 1 by 
a quantity of the order of a 2 . But nevertheless the arcs which subtend these abscissae 
differ from a quadrant of the ellipse by a quantity of the order of a. This arises from 
the direction of the arc at the extremities of the axis being perpendicular to the 
abscissa, so that its increment is infinitely greater than that of the latter. It will be 
well to show the truth of the theorem in this case. When x = u we have (putting 
1 - e 2 = b 2 ) 
For from these values we deduce y -{- z = 0, and thence (neglecting quantities of the 
order cJ 3 ) 
x = 0 arc x = 0 = (1.) 
y = 1 arc y — Q = (2.) 
2 = — 1 arc z = — Q = (3.) 
and the sum, viz. 
(l.)+(2.)-(3.)=2Q, 
8 
x -\ -y-\-z = x — u 
xy z = — - u. 
So that xy z are roots of 
