Lagrange J.-L. Lectures on elementary mathematics (Chicago, 1898)(T)(175s)_M

Lagrange J.-L. Lectures on elementary mathematics (Chicago, 1898)(T)(175s)_M_.djvu

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Date Jul 15, 2005

But if we should attempt to
apply the preceding method to cubic radicals we
should be led again to equations of the third degree
in the irreducible case...
have the same chord, but taken negatively, for on
completing a full circumference the chords become
zero and then negative, and they do not become posi-
positive again until the completion of the second circum-
circumference, as you may readily see...
Let us take again the equation
and let us suppose that its three roots are a, b, с
84 ON ALGEBRA...
But
this generality is needless and prejudicial to the sim-
simplicity which is to be desired in the expression of
the roots of the proposed equation, and we should
prefer the formulae which you have learned in the
principal course and in which the three roots of the
reduced equation are contained in exactly the same
manner...
To this end, instead of sup-
supposing the right-hand side of the equation equal to
zero, we suppose it equal to an undetermined quan-
quantity y...
Whence it follows that
the curve will have a continuous and single course,
and that it may be extended to infinity on both sides
of the origin O...
Conversely, every value of x which is a root of the
equation will be found between some larger and some
RESOLUTION OF NUMERICAL EQUATIONS...
But in the majority of cases it is not sufficient to
know the limits of the roots of an equation; the thing
necessary is to know the values of those roots, at
least as approximately as the conditions of the prob-
problem require...
If we ex-
examine the curve of the equation it will be readily seen
that the question resolves itself into so selecting the
values of x that at least one of them shall fall between
two adjacent intersections, which will be necessarily
the case if the difference between two consecutive val-
RESOLUTION OF NUMERICAL EQUATIONS...
But since each difference can be positive
or negative, it follows that the equation of differences
must have the same roots both in a positive and in a
negative form; that consequently the equation must
be wanting in all terms in which the unknown quan-
quantity is raised to an odd power; so that by taking the
square of the differences as the unknown quantity, this
i i-i i • , m i.m — 1) ,
unknown quantity can occur only in the ——— -th
Li
degree...
of
the proposed equation; so that we may take for M
the quantity which is numerically the greatest of
these...
These two equations will, accordingly, have
a common divisor which can be found by the ordinary
method, and this divisor, put equal to zero, will give
one or several roots of the proposed equation, which
roots will be double or multiple, as is easily apparent
from the preceding theory; for if the last term Q of
the equation in * is zero, it follows that
« = 0 and a — b...
Furthermore, when there
are as many results found as there are units in the
highest exponent of the equation, we can continue
these results as far as we wish by the simple addition
of the first, second, third differences, etc., because
the differences of the order corresponding to the de-
degree of the equation are always constant...
The same construction and the same demonstra-
demonstration hold, whatever be the number of terms in the
proposed equation...
N
or
We will now consider the curve having the equa-
equation
M N
х*+!^~х?-А=У
in which it will be seen at once that by giving to л а
M
very small value, positive or negative, the term —^,
while continuing positive, will grow very large, be-
because a fraction increases in proportion as its denomi-
denominator decreases, and it will be infinite when x = 0...