6 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
If we consider the generality of the supposition [1.], which admits any number of 
arbitrary quantities, it certainly appears remarkable that this equation f. v = 0 should 
be always verified. 
§.3. Application to the Parabola. 
If the tangent at the vertex of a parabola be taken for the axis of abscissae, and the 
semiparameter = 1, then if x be the abscissa, the equation of the curve will be 
2 y = x 2 , 
and the arc, which may be designated as arc x, 
—Jdx ■v'T + x 1 . 
The known value of this is 
[2.] Arc x = ^x\/l-{-x 2 + ^ log (x -f- \/ 1 -{-x 2 ). 
This is a function of x, the properties of which appear to have been hitherto little 
examined. I will establish two theorems concerning it, which are of considerable 
simplicity. 
Theorem I . — If three abscissae are the roots of the equation 
x 3 — r x 2 -f- + 1 ^ x — r = 0, 
the sum of the arcs equals the sum of the abscissae . 
Since each arc is greater than its corresponding abscissa, it is evident that the 
word sum is to be understood in an algebraic sense, or that at least one of the arcs 
must be taken negatively. 
Theorem II . — If three abscissae are the roots of the equation 
x 3 - a x 2 — a b — x + — 0, 
the sum of the arcs equals the product of the abscissae. 
This theorem is remarkable for its simplicity, when it is considered that it contains 
two arbitrary quantities, a and b, which, as it appears, may have any values. 
Demonstration of Theorem I. 
Put J 1 + 3C ■ " ■ X 2 + v x -j- 1 : whence 
f 1 .1 x 3 -f- 2 v x 2 + (v 2 + 1 ) x -f- 2 v — 0. 
Also 
S dx 1 -f- x 2 = Sx 2 dx-\-vSxdx + S d x 
.’. S J* d x \/ \ + x 2 = -f* J* S#. 
S x 3 
KJ tAj 10 
The first term ^ : fr- ~ —■ — p q -f- r in all equations. Here p = — 2 v q = v 2 l 
r — — 2 v. 
