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MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
Attribute to v any numerical value, and let the three roots of the equation then 
be m , m', m". And when v has some other value, let the roots be n, n', n". So that 
while v lias changed progressively from one value to the other, the root m has pro- 
gressively changed its value to n, the root w! to n', and the root m" to n". 
These things being thus understood, the meaning of the theorem is, that the value 
of the integral [ J*d x v^T x + 1 taken between the limits x — m, x — n, 
+ its value between the limits m', n', 
+ its value between the limits m", n", 
= a constant. 
If the question be viewed geometrically, since the roots of an equation are the 
intersections of a curve with its axis, a progressive change in the value of (k— v), the 
absolute term, is equivalent to a displacement of the axis parallel to itself, in conse- 
quence of which all the intersections change their places simultaneously. 
In the case of two variables, we have simply 
X = x 2 — a x b = v, 
or 
x 2 — a x + (b — v) = 0. 
And if x, y, are the roots of this equation, the theorem becomes 
/?X . d x +/ <pY . dy = const., 
<p being any function. 
Now in this particular case the theorem admits of a very simple demonstration. 
For since x y = a, y — a — x; and substituting this value in Y = y 2 — a y + b, 
it becomes (« — x) 2 — a [a — x) b = x 2 — a x b\ also dy becomes — d x. 
.*. <p (y 2 — ay b) dy becomes — <p {pc 2 — a x + b) dx. 
Therefore 
<p {x 2 — a x + b) d x + <p (y 2 — ay -{- b ) dy — 0, 
or 
. d x <pY . d y = 0 
d x - f- f<p Y . dy — const., 
which was to be demonstrated. 
Let X — x n — a j:” - 1 + .... as before, it may be shown upon the same principles 
that S f <p X . x m d x = const., provided Sx m dx = 0, or S x m+ 1 is constant, that is to 
say, does not contain v ; which depends on the relative values of m and n. Also we 
may obtain in a similar manner the solution of the following problem, viz. 
S f <p (Jxj) dx = const., 
where X is a polynomial of n dimensions, and X' another polynomial of not more than 
n — 2 dimensions. For, putting 
