where the ambiguous sign + indicates that the sign may be + or according to the relative magnitudes of the quantities with contrary signs in the partial products, and where it will be observed the permanencies in the proposed equation are changed into signs of ambiguity; hence the permanencies, take the ambiguous sign as you will, are not increased in the final product of the introduction of the positive root+a; but the number of signs is increased by one, and therefore the number of variations must be increased by one. Hence

it is obvious that the introduction of every positive root also introduces one additional variation of sign; and therefore the whole number of positive roots cannot exceed the number of variations of signs in the successive terms of the proposed equation.

Again, by changing the signs of the alternate terms, the roots will be changed from positive to negative, and vice versa (see Prop. V). Hence the permanencies in the proposed equation will be replaced by variations in the changed equation, and the variations in the former by permanencies in the latter; and since the changed equation cannot have a greater number of positive roots than there are variations of signs, the proposed equation cannot have a greater number of negative roots than there are permanencies of signs.

EXAMPLES.

(1.) The equation x6+3x5—41xa—87x3+400x2+444x-720=0, has six real roots. How many are positive?

(2.) The equation x3-3x3-15x2+49x-12=0, has four real roots. How many of these are negative?

TRANSFORMATION OF EQUATIONS.

PROPOSITION I.

To transform an equation into another whose roots shall be the roots of the proposed equation increased or diminished by any given quantity.

.....

Let ax+A+A112+ A-1+A, 0, be an equation, and let it be required to transform it into an equation whose roots shall be the roots of this equation diminished by r.

This transformation might be effected by substituting y+r for x in the proposed equation, and the resulting equation in y would be that required; but this operation is generally very tedious, and we must therefore have recourse to some more simple mode of forming the transformed equation. If we write y+r for x in the proposed equation, it will obviously be an equation of the very same dimensions, and its form will evidently be

ay"+By+B1y"-2+

But y=x-r, and therefore (1) becomes

... B-1y+B=0

B-1(x-r)+B=0

(2);

a (x−r)"+B, (x—r)1-·1+ which, when developed, must be identical with the proposed equation; for, since y+r was substituted for x in the proposed, and then x—r for y in (2), the transformed equation, we must necessarily have reverted to the original equation; hence we have

a(x—r)"+B1(x—r)"'+ . . B2-1(x−r)+B„=ax”+A ̧xa¬'x+ · · „‚×+„. Now if we divide the first member by x-r, the remainder will evidently be B., and the quotient

-1

a(x−r)11+B1(x—r)1-2+ . . . . . B„-2(x−r)+B1-1,; and since the second member is identical with the first, the very same quotient and remainder would arise by dividing this second member also by x-r; hence it appears that if the first member of the original equation be divided by x-r, the remainder will be the last or absolute term of the sought transformed equation.

Again, if we divide the quotient thus obtained, viz..

a(x-r)"'+B1(x−r)"¬2+ ·

by x-r, the remainder will obviously be B-1, the coefficient of the term last but one in the transformed equation; and thus by successive divisions of the polynomial in the first member of the proposed equation by x-r, we shall obtain the whole of the coefficients of the required equation.

RULE.

Let the polynomial in the first member of the proposed equation be a function of x, and r the quantity by which the roots of the equation are to be diminished or increased; then divide the proposed polynomial by x-r, or x+r, according as the roots of the proposed are to be diminished or increased, and the quotient thus obtained by the same divisor, giving a second quotient, which divide by the same divisor, and so on till the division terminates; then will the coefficients of the transformed equation, beginning with the highest power of the unknown quantity, be the coefficient of the highest power of the unknown in the proposed equation, and the several remainders arising from the successive divisions taken in a reverse order, the first remainder being the last or absolute term in the required transformed equation.

Note. When there is an absent term in the equation, its place must be supplied with a cipher.

EXAMPLES.

(1.) Transform the equation 5xa—12x3+3x2+4x-5=0 into another whose roots shall be less than those of the proposed equation by 2.

x-2)5x1—12x3+3x2+4x-5(5x”—2x2−x+2

This laborious operation can be avoided by Horner's Synthetic Method of division; and its great superiority over the usual method will be at once apparent by comparing the subsequent elegant process with the work above. Taking the same example, and writing the modified or changed term of the divisor -2 on the right hand instead of the left, the whole of the work will be thus arranged:

.. 5y+28y3+51y2+32y-1=0 is the required equation, as before. (2.) Transform the equation 5y1+28y3+51y2+32y—1=0 into another having its roots greater by 2 than those of the proposed equation.

.. 5x1—12x3-3x2+4x-5=0 is the sought equation; which, from the transformations we have made, must be the original equation in Example 1.

(3.) Find the equation whose roots are less by 1-7 than those of the equation z3—2x2+3x-4=0.

Now we know the equation whose roots are less by 1 than those of the given equation: it is x23+x2+2x-2=0; and by a similar process for ⚫7, remembering the localities of the decimals, we have the required equation; thus:

••• y3+3·ly2+4.87y+233=0 is the required equation.

This latter operation can be continued from the former, without arranging the coefficients anew in a horizontal line, recourse being had to this second operation merely to show the several steps in the transformation, and to point out the equations at each step of the successive diminutions of the roots. Combining these two operations, then, we have the subsequent arrangement.

We have then the same resulting equation as before, and in the latter of these we have used 17 at once. It is always better, however, to reduce continuously as in the former, to avoid mistakes incident to the multiplier 1·7. (4.) Find the equation whose roots shall be less by 1 than those of the equation

x3—7x+7=0.

(5.) Find the equation whose roots shall be less by 3 than the roots of the equation

x-3x3-15x2+49x-12=0,

and transform the resulting equation into another whose roots shall be greater

by 4.

(6.) Give the equation whose roots shall be less by 10 than the roots of the equation

x1+2x2+3x2+4x-12340=0.

(7.) Give the equation whose roots shall be less by 2 than those of the equation

x+2x3—6x2-10x+8=0.

(8.) Give the equation whose roots shall each be less by than the roots of the equation

To transform an equation into another whose second term shall be removed. Let the proposed equation be

and by Prop. IV. we know that the sum of the roots of this equation is — A1; therefore the sum of all the roots must be increased by A,, in order that the transformed equation may want its second term; but there are n roots, and A1 hence each root must be increased by A, and then the changed equation will

have its second term absent. If the sign of the second term of the proposed equation be negative, then the sum of all the roots is +A,; and in this case we must evidently diminish each root by and the changed equation will then have its second term entirely removed. Hence this

Rule. Find the quotient of the coefficient of the second term of the equation divided by the highest power of the unknown quantity, and decrease or increase the roots of the equation by this quotient, according as the sign of the second term is negative or positive.

EXAMPLES.

(1.) Transform the equation ∞3—6x2+8x—2—0 into another whose second term shall be absent.

Here A, =—6, and n=3; .. we must diminish each root by g or 2.

.. y3-4y-2-0 is the changed equation.

And since the roots are diminished we must have the relation x=y+2.