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Stern's Theorem.

of

Find the row of signs corresponding to the values

u, U, U', U", &c.,

for any value p of the variable, and also for a value q of the variable.

The number of real roots of the equation, comprised between p and q, cannot be greater than the difference between the number of permanences of the first row of signs and that of the second row.

Proof. First. It may be shown as in the third division of the proof of art. 290, that one permanence at least is always lost from the row of signs when the variable in decreasing passes through a value which is one of the roots of the equation.

Secondly. When any term of the series except the first or the last, vanishes, it passes, by art. 288, with the decreasing variable, from having the same sign with its derivative, which is the next term of the series, to having the reverse sign of it. Even then, if it had before the change the reverse sign of the preceding term and after the change the same sign, it introduces a permanence which is only sufficient to take the place of the other permanence which is lost. The number of permanences of the row of signs is not, therefore, augmented by the vanishing of such an intermediate term.

Thirdly. The last term of the series must be constant, for the number of dimensions is diminished by each derivation; and, therefore, as z decreases from a value p to a smaller value q, the number of permanences of the row of signs must be diminished by as large a number at least as the number of roots comprised between p and

Descartes' Theorem.

295. Definition. An equation

x2 + a x2-1+ &c. + k x2 + k x + l=0. is said to be complete in its form, when it contains terms multiplied by every different power of x from the highest to unity, and also a constant term, such as 1.

296. Descartes' Theorem. A complete equation cannot have a greater number of positive roots than there are variations in the row of signs of its terms, nor a greater number of positive roots than there are permanences in this row of signs.

Proof. If the equation is that of art. 295, the values of u, U, U', &c. in art. 294, are

u = x2 + a xn−1 + &c. + h x2 + k x + l

U = nxn−1 + (n − 1) a x2-2+ &c.+2hx+k
U' = n (n−1) xn−2+(n−1)(n-2)ax"¬3+&c.+2h

&c.

The row of signs when x = ∞ is

++++ +, &c.,

consisting wholly of permanences.

When x- ∞ it is

±, F, ±, F, &c.,

in which the upper row of signs is used when ʼn is even,

and the lower row when n is odd.

consists wholly of variations.

The row of signs when x = 0 is

In either case, this row

(like ),(like k), ±(like 2 h)

that is, it is the same as the row of signs formed by the

terms of the equation taken in the inverse order.

Zero substituted for a Term which is wanting.

The limit of the number of positive roots is, therefore, by art. 294, equal to the excess of the whole number of pairs of successive signs of the terms, over the number of permanences; that is, it is equal to the number of variations; and in the same way, the number of negative roots cannot exceed the number of permanences.

297. Corollary. The whole number of successions of signs of an equation, that is, the sum of the permanences and variations, is one less than the number of terms, or the same as the degree of the equation, that is, the same as the number of roots.

If, therefore, all the roots are real, the number of positive roots must be the same as the number of variations, and the number of negative roots must be the same as the number of permanences.

298. Scholium. Whenever a term is wanting in an equation, its place may be supplied by zero, and either sign may be prefixed.

299. Corollary. When the substitution of +0 for a term which is wanting gives a different number of permanences from that which is obtained by the substitution of -0, and consequently a different number of variations also, the equation must have imaginary roots.

300. Theorem. When the sign of the term which precedes a deficient term is the same with that which follows it, the equation must have imaginary roots.

Proof. For if the terms which precede and follow the deficient term are both positive, the substitution of +0 gives two permanences; while the substitution of 0 gives

Imaginary Roots, when Terms are wanting.

two variations.

The reverse is the case when both these terms are negative. The equation must, therefore, in either case, have imaginary roots.

301. Theorem. When two or more successive terms of an equation are wanting, the equation must have imaginary roots.

Proof. For the second deficient term may be supplied with zero affected by the same sign as that of the term preceding the deficient terms; and the first deficient term is then preceded and followed by terms having the same sign, so that there must, by the preceding article, be imaginary

roots.

302. Theorem. When an uneven number (m) of successive terms is wanting in an equation, the number of imaginary roots must be at least as great as (m + 1), if the term preceding the deficient terms has the same sign with the term following them; and the number of imaginary roots must be at least as great as (m 1), if the term preceding the deficient terms has the reverse sign of the term following them.

Proof. First. If the sign of the term preceding the deficient terms is the same with the sign of the term following them; supply the place of each deficient term with zero affected by this same sign. All the (m+1) successions, dependent upon the deficient terms, must in this case be permanences. But if the sign of every other zero beginning with the first is reversed, namely, of the first, third, fifth, &c., all these permanences are changed into variations; so that (m+1) roots can be neither positive nor negative, and are, consequently, imaginary.

Superior Limit of Positive Roots.

Secondly. If the sign of the term preceding the deficient terms is the reverse of the sign of the term following them; supply the place of the two last deficient terms with zero affected by the same sign as that of the term preceding the deficient terms. This case becomes the same as the preceding with (m· -2) deficient terms, and there must therefore be (m-1) imaginary roots.

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303. Theorem. When an even number of successive terms is wanting in an equation, the number of imaginary roots must be at least as great as the number of these deficient terms.

Proof. Let the place of the first deficient term be supplied by zero affected with the same sign as that of the term which follows the deficient terms.

The number of deficient terms is thus reduced to the uneven number m—1; and, as the term preceding the deficient terms is now of the same sign with that of the term following them, the number of imaginary roots of the equation must, by the preceding article, be at least as great

as

(m1)+1=m.

304. A number, which is greater than the greatest of the positive roots of an equation, is called a superior limit of the positive roots; and one, which is less than the least of the positive roots, is called an inferior limit of the positive roots.

In the same way, a superior limit of the negative roots is a number which, neglecting the signs, is greater than the greatest negative root; and an inferior limit of the negative roots is a number which is less than the least negative root.

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