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br-1 = -01;
6,-2= - }(6-10, +0,);
br-3 = - }(br-20, +6,-10, +03);
6,-43 -1 (6r-30, +br-20, +6,-103 +0.);

&c. = &c.

n(n-1)
1. 2

If the given equation is a cubic, the transformed equation is also a cubic, and

b,=2(3a, – až);
b = (3a, -a)";
b =-

} {4(3a, – ai)(a 3a,.a) + (9a-ajan)}. If the given equation is a biquadratic, the transformed equation is of six dimensions, and bg =8a, ; b4=22 až +8a; bg = 18a} – 16 a,.a – 26 a;; b,= 17az + 24 ağ.a7.24.a’ +3.24.az.a ; 6=4 a +2.33.az.až +23.39.a .a-3.2o.az.a? + 2'.az.a; b = 28.a’ – 27.az.a? + 24.3.a,.a.a +24.az.a - 22.a.a; - 3.a.

The biquadratic is here supposed to have been deprived of its 2nd term.

If b is negative the biquadratic will have two possible, and two impossible roots: if b is positive, the roots are all possible, when b, and b4 are positive, and , bz, b5, negative; otherwise they are all impossible.

To obtain the equation of differences by elimination : for X substitute w+x in (x)=0, and in after suppressing («) in the transformed equation, (Art. 33) eliminate x between that equation, and px)=0. If y be substituted for you in the resulting equation, which contains only the even powers of x, it will appear under the same form as above.

(G. 369—9; G. A. Chap. vi; Bour. 310, 1.)

LIMITS OF THE ROOTS OF EQUATIONS.

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(40.) The limits of the roots of an equation, if substituted successively for the unknown quantity, give results alternately positive, and negative.

If two quantities when substituted in an equation give results having different signs, an odd number of roots must lie between them: if the results have the same sign, an even number of roots, or none, must lie between the quantities substituted.

To find a superior limit to the roots of an equation ; in the transformed equation, Ply+c)=0, assume

C such that all the coefficients may be positive; this value of c will be a superior limit.

To find an inferior limit, change the signs of all the roots, and proceed as before.

The greatest negative coefficient increased by unity is a superior limit.

If an-, is the first, and an-r the greatest negative coefficient, then

1

(a,
'n-,)+ 1 is a superior limit.

(Maiziere, Ann. de Math. Tome 3.)

A negative coefficient less than the greatest increased by unity will frequently be found to be a superior limit: let Qp be the greatest negative coefficient, and a +1 positive, then since

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this part of the equation is essentially positive for all values

a.

therefore the next greatest negative coefficient a +1

of 3 >

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If the roots are all possible, and positive, each of the following quantities is a superior limit;

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The roots of the equation Rw)=0 are limits between those of the equation P(x)=0, when the roots of the latter are possible.

If all the roots of an equation are positive, or all negative and its terms be multiplied by the terms of any arithmetical progression, the resulting equation will be a limiting equation to the former,

If the equation has both positive and negative roots, the same is true; with this exception, that either two of its roots or none lie between the positive and negative roots of the original equation, according as a decreasing or an increasing progression is used.

(W. 298–318; Bour. 318-27; F. 507–11.)

G

Descartes's Rule. An equation cannot have a greater number of positive roots, than it has changes of sign from term to term, nor a greater number of negative roots than it has continuations of the same sign.

If the roots are all possible, there are exactly as many of them positive as there are changes, and as many negative as there are continuations of sign.

(Bour. 328–30; G. Chap. 34.)

INVESTIGATION OF COMMENSURABLE Roots.

(41.) Let a be a commensurable root of p(x)=0, and therefore an integer ; and assume

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919929 &c. 9n-1; are all integers. Hence, to find the commensurable roots: write down in a horizontal line all the divisors of the last term, negative as well as positive, that are between the nearest assignable limits of the greatest positive, and negative roots; under these write the corresponding quotients of the last term divided by each of them; add to each of these the coefficient of X, and write down the sum under the corresponding quotient; then divide each of these sums by the divisor standing over it, and write down the quotient under it, neglecting all fractional quotients; to the remaining quotients add the coefficient of , and proceed as before; and so on : those divisors which give always an integral quotient, will be found to be roots of the equation ; which may therefore be reduced as many degrees.

If there are equal roots, this method will only detect one of them, it should therefore be repeated with the resulting equation.

We need not include 1 or – 1 amongst the divisors, as it is easy to ascertain whether either is a root of the equation.

The number of divisors to which the above process is applied may be thus reduced. Let by, b, be the results obtained by substituting +1 and – 1, in the equation :

[1] Every positive divisor which when diminished by unity does not divide bı, and when increased by unity, does not divide bą, may be rejected.

[2] Every negative divisor which when increased by unity does not divide bı, and when diminished by unity does not divide bı, may also be rejected.

(Bour. 331-7; L. 199–203.)

If three, or more terms of the arithmetical progression 1, 0, -1, &c. be substituted for the unknown quantity, and the divisors of the results, taken in order, be formed into arithmetical progressions, those divisors of the last term which occur in these progressions may be found on trial to be roots of the equation.

(W. 336-40.)

INVESTIGATION OF INCOMMENSURABLE Roots.

Suppose the difference between any two

(42.) Case 1st. possible roots >1.

[1] Lagrange's Method. Find a number b, such that one of the

1 roots is >b, but <b, +1, and for a substitute bi + - in the

Ny given equation. The resulting equation in x, must have at least one possible root > 1, let this root be >b, and < b +1;

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