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from what has gone before, we shall likewise have
у = m ^A or у = n
and the same will also hold good for z...
Extracting
the square root, so as to obtain the equivalent expres-
expression we have
sion, we have
for the real value of the imaginary quantity we started
with...
But how is this value of x to be assigned? It would
seem that it can be represented only by an imaginary
expression or by a series which is the development of
an imaginary expression...
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 for the rest, if we make u^=s—?- in the re-
b
duced equation in u, so as to eliminate the second
term and to reduce it to the form which we have above
ON ALGEBRA...
Therefore, there must of necessity be some
expression for the value of x between A and В which
will make />-—<2; just as two moving bodies which
we suppose to be travelling along the same straight
line and which having started simultaneously from
two different points arrive simultaneously at two other
points but in such wise that the body which was at first
in the rear is now in advance of the other,—just as
two such bodies, I say, must necessarily meet at some
RESOLUTION OF NUMERICAL EQUATIONS...
But there is a more general and simpler method
Application of considering equations, which enjoys the advantage
°o ХёьгаУ °* affording direct demonstration to the eye of the
principal properties of equations...
tion, above and below the axis, points of the curve are
situated as are the points L, Q with respect to the in-
imersec- tersection M...
It is also
easy to see that there can be given to л a positive or
negative value sufficiently great to make the first term
xm of the equation exceed the sum of all the other
terms which have the opposite sign to x'"; with the
result that the corresponding value of у will have the
same sign as the first term x"'...
But if there are several positive terms in the equa-
equation preceding the first negative term, we may take
for k a quantity less than the greatest negative coeffi-
coefficient...
We can
then by new substitutions bring these two limits still
closer together and approach as nearly as we wish to
the roots sought...
And we must then seek, by the methods expounded
above, a quantity smaller than the smallest root of
this last equation, and take that quantity for the value
of D...
Let
the greatest of the numerical quantities obtained in
The arith- this manner be called M...
The point Q where this straight line cuts the perpen-
perpendicular PT v/ill give the segment PQ=y...
The method
which I last expounded to you for finding and demon-
demonstrating divers general properties of equations by con-
considering the curves which represent them, is, properly
speaking, a species of application of geometry to al-
algebra, and since this method has extended applica-
cations, and is capable of readily solving problems
whose direct solution would be extremely difficult or
even impossible, I deem it proper to engage your at-
attention in this lecture with a further view of this sub-
128 THE EMPLOYMENT OF CURVES...
Let a be the distance between the two lights and
x the distance between the point sought and one of
the lights, the intensity of which at unit distance is
M, the intensity of the other at that distance being
N...
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