Lagrange J. L. Лекции по элементарной математике (Чикаго, 1898) М. , Lagrange J.-L. Lectures on elementary mathematics (Chicago, 1898)(T)(175s)_M

Lagrange J. L. Лекции по элементарной математике (Чикаго, 1898) М.

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

Size 0.8Mb
Date Jul 15, 2005

Cites: Therefore, if the value of л'3 be increased by
insensible degrees from 8/ to infinity, the value of g*
will also augment by insensible and corresponding
degrees from zero to infinity...
Although the preceding discussion may be deemed
sufficient to dispel all doubts concerning the nature
of the roots of equations of the third degree, we pro-
propose adding to it a few reflexions concerning the
method by which the roots are found...
As the degree of the equation is odd we may be cer-
certain, as you doubtless already know and in any event
will clearly see from the lecture which is to follow,
that it has necessarily one real root...
And since the radical
V—3 may also be taken negatively, we shall also
have the expression
1 _|_ 1/H3 l _ i/—3
b
_ b a
expressed in known quantities, from which the values
of a and b can be deduced...
Hence the product of the three quantities jc,
8
z, t, that is to say of the three radicals
V a, V'b, V'c,
must have the contrary sign to that of the quantity q...
I propose here to set forth the principal artifices
which have been devised for accomplishing this im-
important object...
having been described in the manner indicated, it is
clear that its intersections with the axis AB will give
the roots of the proposed equation
For seeing that this equation is realised only when in
the equation of the curve у becomes zero, therefore
those values of x which satisfy the equation in ques-
question and which are its roots can only be the abscissas
that correspond to the points at which the ordinates
are zero, that is, to the points at which the curve cuts
the axis AB...
For the same reason there can be no simple inter-
intersection unless on both sides of the point of intersec-
Ю4 RESOLUTION OF NUMERICAL EQUATIONS...
shall offer a very simple demonstration of it,—a dem-
demonstration which will enjoy the collateral advantage
of furnishing a limit beyond which we may be certain
no root of the equation can be found...
^ a vajue such that the coefficients of all the terms
become positive, it is plain that there will then be no
positive value of z that can satisfy the equation...
Then
one of the two successive values of x which have given
results with contrary signs will necessarily be larger
than the root sought, and the other smaller ; and since
by hypothesis these values differ from one another
only by the quantity n, it follows that each of them
approaches to within less than n of the root sought,
and that the error is therefore less than n...
For example, if m
is the degree of the original equation, that of the equa-
equation of differences will be m(m — 1), because each root
can be subtracted from all the remaining roots, the
number of which is m—1,—which gives m(m — 1)
differences...
Put-
Putting, then,
f=zVk + l and g=fyh + l,
we shall have f and —g for the limits of the positive
and negative roots...
Since, however, the finding of the equation in у by
General the ordinary methods of elimination may be fraught
formulae jth considerable difficulty, I here give the general
for ehmina...
Then taking the unknown
quantity for the abscissa x, and the function of the
unknown quantity, or the quantity compounded of
that quantity and the known quantities, which forms
one side of the equation, for the ordinatejc, the curve
described by these co-ordinates x and у will give by
its intersections with the axis those values of x which
are the required roots of the equation...