The same result as was obtained by the direct solution of

the above equation under Art. 151.

2. Transform the cubic equation

2

x2+A1x2+A2x+A3=0,

into a new equation wanting its second term.

We might proceed in this way for the transformation of equations of higher degrees, but its easy to see that this method would be very lengthy and complicated for such equations, we shall therefore seek some law by which these transformations can be made with less labor.

we substitute x'+u for x, and imitate the operations of Art. 255, we shall have

If, in the above transformation, we put X for the coefficient of u', or which is the same thing for the terms independent of u.

Also, put X" for the coefficient of u, and X

X for the coefficient of u?, 2

for the coefficient of us,

2.3 and so on, we shall have

3

n

in

An-1)

X=2*+A12-1+A221-2---- An-13+ A,
X'=x'"+Ax'n-1 + A22'n–2 An-12' + Ang
X"=nx'n-1 +(n-1)A12'n–2+(1-2) A2x'n-3
X"'=n(n-1)x'n-2 +(n-1)(n-2)Az x'n=3+
X!!=n(n-1)(n—2) xun–3+(n-1)(n-2)(n-3)A4%-6...
&c.,

r

If we examine the above expressions, we shall discover the following law:

X' is derived from the general equation X, by simply changing x into x'.

X" is derived from X' by multiplying each of the terms of X' by the exponent of x' in that term, and diminishing this exponent by a unit.

X" is derived from X" in the same manner as X' was derived from X'.

And in general, a coefficient of any rank, in the above transformed equation, is formed by means of the preceding, by multiplying each term of the preceding by its exponent, and dividing the product by the number of coefficients which precedes the terms sought, and diminishing the exponent by a unit.

(258.) The polynomial X" is called the first derived polynomial of X'.

The polynomial of X"" is called the second derived polynomial of X' and so on for the succeeding polynomials.

(259.) We will add a few examples to illustrate the above law.

1. Transform the equation

x-12x+17x2-9x+7=0,

into an equation wanting is second term.

12

By Art. 256, we must substitute u+ -=u+3 or 3+u for

4

x, this transformed by Art. 257, will be of this form

X' (3)^-12(3)3+17(3)2-9(3)1+7=-110,

X"=4(3)3—36(3)2+34(3)1—9

x5-10x+7x3+4x-9=0,

into a new equation wanting its second term.

Proceeding as above, we find

3

=-123,

X'=(2)-10(2)+7(2)+4(2)1-9=- 73,

Hence, our transformed equation is

u5-33u3-118u2-152u-73=0.

3. Transform

3x+15x2+25x-3=0,

into an equation wanting its second term.

Dividing each term by 3, in order to make it

Now, in order to make the second term disappear, we

In this example, the third term vanished at the same time

as the second.

4. Transform

3

4x3-5x2+7x-9=0,

into a new equation of which the roots shall exceed by a unit each of the corresponding roots of the given equation.

We must assume u=x+1 or x=u−1, which gives

X'=4(-1)3-5(−1)2+7(−1)1-9=-25,

X" 12(-1)2-10(-1)1+7

= 29,

=-17,