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+0.0061+ etc.; and taking the upper and lower signs in the ambiguous signs separately, we have 1.880 + and −1.535 for the first figures of the two remaining roots, the first of these being correct to two and very nearly to three places of figures, and the second being correct to three places of figures. It is manifest that the preceding method in the present question is much more simple than to find the first figures by Sturm's theorem.

2. To find the number of real roots contained in 3x1 12x2+8x - 24 0.


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Here the first derived function may be expressed by 336x+2, and thence we deduce a2 - +4 for the first modified remainder; and since the modified remainder is clearly positive for all real values of x, it is evidently unnecessary to get any more of the derived functions. Hence (4) may be represented by 3x-12x2+ 8x - 24, 3x2-6x + 2, x2 — x +4; and if in these we put co and co for x, they may be expressed by 3 ∞)1, — 3( ∞)3, (), two variations, and 3(∞)1, 3(∞)3, (∞)2, no variation; consequently, the given equation has but two real roots.

If -3 is put for x, the signs are +, -, +, two variations;

2 and

and if 2 is put for x, the signs are —, —, +, one variation; consequently, one of the roots lies between -3, and 2 is the principal part of the root.

In like manner, we easily get 2 for the principal part of the remaining root.

3. To find the number of real roots of a - 2x22 + 6x2 + 24. Here the first derived function is (4x-3x + 12)x, or, since 4x-3x+12 is positive for all values of x, it may be rejected, and we may take x for the first derived function.

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Dividing a 2+6x+ 24 by x, we get 24 for the remainder, whose sign, being changed, gives - 24 for the modified remainder; consequently, (4) become a — x3 + 6x2 +24, x, 24.

Putting -∞ and ∞ for x in the preceding functions, they may be expressed by (∞), -∞, 24, one variation, and (∞), ∞,- 24, one variation; consequently, the proposed equation has no real roots.

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4. To find the number of real roots of x2 + 8x2 + 17x2 бас 36 = 0.


Here 2x+12+17x-3 may be taken for the first derived function, and thence we get 7x2 +43 +66 for the first modified remainder, and +3 is the second and last modified remainder; consequently, the given equation has equal roots.

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∞ and

Hence, (4) become a +8 +17x2-6x-36, 2x2 + 12x2 + 17x-3, 7x2 + 43x + 66, x+3; and putting for, these become (), -2(∞)3, 7(∞)2, ∞, three variations, and (∞), 2(∞), 7(∞), o, no variation; consequently, the equation has three unequal real roots; indeed, the equation has a pair of equal roots, whose common value - 3, and two other roots, one of which lies between -3.3 3.2, and the other between 1.2 and 1.3.


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5. To find the number of real roots of 2x1- 20x + 19 = 0. Here the first derived function is 8-20, or, rejecting the useless factor 4, it may be represented by 2x3 — 5.

Dividing 220x+19 by 2-5, we get 15x + 19 for the remainder, whose signs being changed, give 15a 19 for the first modified remainder.


Multiplying 2-5 by 15 × 15, we get 450-1125 for the product, which, divided by 15x-19, gives

the remainder, whose sign being changed, gives





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second modified remainder, which may clearly be expressed by 3157; hence, (4) become 24 - 20x + 19, 2x3- 5, 15x - 19, 3157.


Because the proposed equation (clearly) has no negative roots, and that the signs of the preceding functions are the same for x= O that they are when a = 1, we put x = 1, and the signs are +, -, -, +, two variations; and putting a = 2, the signs are +, +, +, +, no variation; consequently, the given equation has only two real roots, and they lie between 1 and 2.

6. To find the number of real roots of 3 + 11x2 - 102x + 181 = 0.

Here 3+22-102 is the first derived function; and dividing 9 times the given equation by it, we get 8542751 for the first modified remainder; or, rejecting the fac tor 7, it may be expressed by 122x-393.

Multiplying the first derived function by (122), and dividing the product by 122x - 393, we get — (122)2 × 102 +3863 × 3939 for the remainder, whose sign being changed, gives 9 for the second modified remainder.

Since the preceding remainder is so small, it is plain that 122x393 is nearly a factor of the given equation; consequently, putting it equal to 0, we get x 3.221 nearly for one of two roots of the given equation that are nearly equal to each other.

Collecting the functions, we shall get a + 112 - 102x + 181, 3x2+22x-102, 122x-393, 9, for the representatives of (4).

To simplify the preceding expressions, we put x = x′ + 3, and they become a3+20x29x+1, 3x2 + 40x' — 9, 122x-27, 9.

Putting 0.2 and 0.3 for x', the signs of the functions become +, -, -, +, two variations, and +, +, +, +, no variation; consequently, the equation + 20' — 9x+1 =0 has a pair of nearly equal roots, which lie between 0.2 and 0.3, whose first figures are easily found to be 0.213 and 0.229.

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Since + 3, we get 3.213 and 3.229 for the nearly equal roots of the proposed equation; and since 3.221, before found, must clearly lie between these roots, and differ but little from them, it is clear that they can readily be found without using Sturm's theorem.


Putting the equation in 'under the form a' - 9x'-1 + 220, then developing by (c) of Sec. XV., we get

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x' = — 20.44264839, which is correct to six decimal places; consequently, since x=x+3, we get x

17.442648 for one of the roots of the proposed equation, which is correct

in all its figures.

Remark. This example was given by Sturm, in his Memoir, to show the use of his theorem in separating the nearly equal roots of an equation from each other.








(1.) LET Ay" + А1у"-1 + А2у"-2 + Asy-3 + .... + A-13 +A, 0, (1), stand for the equation; then if a' represents a near value of one of the roots and a' + x its exact value, by putting a + for y, we change (1) to A(a' + x)” + A1(a' + x)n−1 + ¿(a' + x)”−2 + . . . . + An-1(a' + x) + A„ = 0, (2). If we put Aa'" + A1a'n¬1 + Â1⁄2α'n-2 + etc. =ƒ(a') = a function of a', and the first derived function of fa'), equal to f'(a), and that of f'(a') equal to ƒ"(a'), and so on, to any required extent; then by expanding the powers in (2), and ordering the result according to the ascending powers of x, we shall get ƒ(a') +ƒ'(a)x +ƒ""(a)2 +ƒ"" (α) + etc. = 0,







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If we divide the terms of (3) by f'(a'), and put f(a')

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÷f(a)=b, and so on, then (3)

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can easily be reduced to x + ax2 + bx3 + cx2 + etc. = u, (a). If we revert the series (a) (see On the Reversion of Series, p. 435), we shall get x = u (b — 2a3)u3 — (c + 5a3 5ab)u* — etc., (b); which will enable us to find a to any degree of exactness after we have found a suitable value of a', and thence the value of the root a' + will be found as required.

To obtain a suitable value of a', we shall put a' = a1 + b1 -1, and a' will be expressed in the proper form (on the supposition that a and b, are not imaginary), for if a' is real, by putting b1 = 0 we shall have a' = a, and if a' is imaginary, then, as has (heretofore) been shown, a + b1 √ — 1 is the proper form for a'; noticing that a, may be 0 if required. Putting a+b-1 for a' in Aa'" + Â ̧a'n¬1 + Å‚a'”—2+


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Let r, r1, r2, rs, . ...n-1 stand for the n roots of (1), whether they are positive or negative, real or imaginary; then, since (1) is equivalent to the product A(y—r). (y — r1). (y — r2) • (y — r3) × . . . . × (y — în−1) = 0, if we substitute a' + for y in this, we shall clearly get f(a') = A(a' — r). (a' — r1). (a' × (a' — r'n−1), (e), and ƒ’(a') = A[(a' — r1). (a' r2). (α' — r3) × .... × (a' — rn-1) + (a' — r). (a' — r2). (a' 13) X.... × (a' — rn-1) + (a' — r). (a' — r1). (a' —rs). (a' r1) × .... × (a'rn-1) + etc.], (f).



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If no two of the roots r, 71, 7, etc., are equal to each other, and we assume a very nearly equal to any one of them, as to r, then since a' -r is a factor of all the terms of (ƒ) but the first, if we omit them on account of their comparative minuteness, it is clear that (f) will give f'(a) = A(a' — r1). (a' — r2). (a' — r'3) × . . . . × (a' —rn-1) nearly; consequently, we shall get u — — - (a′ — r) very nearly, which is very small, on account of the supposed minuteness of a' - r. Hence, if the roots of (1) are unequal to each other, we can assume a real or imaginary value for a', such that it shall differ so little from any real or imaginary root of (1) that the corresponding value of u shall be very small; and it is clear that fla') will (at the same time) also be very small.

f(a) f(a')


If two or more of the roots r, r1, r2, rs, etc., are equal to each other, it is clear that fa') and f'(a') will have a common divisor; consequently, if we divide ƒ(a') and f'(a') by their common divisor, and represent the quotients by F(a') and f(a) F(a) from which F'(a'), we shall have u == f'(a') F'(a') we can find the unequal roots of (1), and that whether they are real or imaginary; it may be added, that if we put the common divisor equal to 0, the values of a', which satisfy the resulting equation, will clearly give the equal roots of (1), after the number of each kind is increased by unity.

(2.) If the root a' + x of (1) is real, we can always find a' to any required degree of exactness by Sturm's theorem; and it is clear that a sufficiently near value of a' can often be

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