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3. When the coefficient of the fourth term divided by that of the second term, gives for a quotient the square root of the last term: then to complete the square, add the square of half the coefficient of the second term, to twice the square root of the last term, multiply the sum by 2, from the product take the third term, and add the remainder to both sides of the biquadratic.

4. The fourth term will be made to go out by the usual operation for taking away the second term, when the difference between the cube of half the coefficient of the second term and half the product of the coefficients of the second and third term, is equal to the coefficient of the fourth term.

IV. Euler's Rule for the Solution of Biquadratics.

Let x-ax3 — bx c = O, be the given biquadratic equa. tion wanting the second term. Take fa, g = a2 +ic, and h = +b2, or √ h = b; with which values of f, g, h, form the cubic equation 23-fz2 + gz h = 0. Find the roots of this cubic equation, and let them be called p, q, r, then shall the four roots of the proposed biquadratic be these following: viz.

When bis positive :

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When b is negative :

1. x = √p + √q + √r. x= √p + √9−√r. 2.x= √ √ √r. | x= √p−√q+ √r. 3. x = −√p + √q−√r. | x = - √p + √9+ √r. 4. x = −√p+ √q + √r. | x=√pqr. Note. 1. In any biquadratic equation having all its terms, if of the square of the coefficient of the 2d term be greater than the product of the coefficients of the 1st and 3d terms, or of the square of the coefficient of the 4th term be greater than the product of the coefficients of the 3d and 5th terms, or of the square of the coefficient of the 3d term greater than the product of the coefficients of the 2d and 4th terms; then all the roots of that equation will be real and unequal : but if either of the said parts of those squares be less than either of those products, the equation will have imaginary

roots.

2. In a biquadratic x+ax + bx2 + cx + d = 0, of which two roots are impossible, and d an affirmative quantity, then the two possible roots will be both negative, or both affirmative, according as a3-4ub + 8c, is an affirmative or a negative quantity, if the signs of the coefficients, a, b, c, d, are neither all affirmative, nor alternately and *.

Various general rules for the solution of equations have been given by Demoivre, Bezout, Legrange, Atkinson, Horner, Holdred, &c.; but the most universal in their application are approximating rules, of which

EXAMPLES.

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Ex. 1. Find the roots of the equation x2 +

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5

This value of x, viz. 2941176, is nearly equal to. To find

17

whether that is the exact root, take the arithmetical comple. ment of the last logarithm, viz. 0.5314379, and consider it as the logarithm of the denominator of a fraction whose numerator is unity thus is the fraction found to be 1 exactly, and this is manifestly equal to 17. 'As to the other root of

5

1695

5

339

748

the equation, it is equal to 12716 17

Ex. 2. Find the roots of the cubic equation

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34

23

441

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4p3>27q2: so that the example falls under the irreducible case.

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a very simple and useful one is given in our first volume. See also J. R. Young's Algebra.

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1612

1323'

x = — sin (60°+^) √!!

The logarithmic computation is subjoined.
Log 1612 3·2073650

Arith. com. log 1323 6.8784402

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half sum = 0.0429026 const. log.

Arith. com. const. log =9.9570974

log 414..

=2.6170003

Arith. com. log 403.7.3946950

log sin 3A = 9·9687927 = log sin 68° 32′ 18′′}.

...

Log sin A
const. log

9.5891206

0.0429026

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=0.0429026

3. sum-10 log-x=0·0395086=log 1·095238=log}}. So that the three roots are,, and

of which the first two are together equal to the third with its sign changed, as they ought to be.

Ex. 3. Find the roots of the biquadratic x1 — 25x2 + 60x-36-0, by Euler's rule.

Here a=25, b=-60, and c=36; therefore

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the square roots of these are √p = √¶ 2 or 4, √r = §. Hence, as the value of 16 is negative, the four roots are

1st x =

1,

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Ex. 4. Produce a quadratic equation whose roots shall be

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Er. 5. Produce a cubic equation whose roots shall be 2, 5, and Ans. 3-4r2-11x+30=0.

Er. 6. roots 1, 4,

3.

Produce a biquadratic which shall have for the 5, and 6 respectively.

Ans. x-6x — 2112 + 146c

Ex. 7. Find r, when x2 + 347x=22110.

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Ex. 8. Find the roots of the quadratic x2

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Ans. x = 10, x =

12

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481860, to find r.

Ans. x=20, x=24093. equation x-3x — 1=0.

Ex. 11. Find the roots of the
Ans. The roots are sine 70°, sin 50°, and

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sin 10°, to a

radius 2; or the roots are twice the sines of those arcs as given in the tables.

Ex. 12. Find the real root of x3-x-6=0.

Ans. √3Xsec 54°44′20′′.

Er. 13. Find the real root of 25x3+75r-46=0.

Er. 14. Given x4 8x3 find x by quadraties.

Ex. 15. Given x1+36x3. to find by quadratics.

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400x3168x + 7744 0, Ans. x=11+ √ 209,

Ex. 16. Given x1+24x3-114x2—24r+1=0, to find x. Ans. x=±√197 −14, x=2±√/5.

Ex. 17. Find x, when x-12x-5=0.

Ans. 1/2, x=-1±2√-L

Ex, 18. Find x, when r1-12x3+47x2-72x+36=0.

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Ans. x = 1, or 2, or 3, or 6.

80a2x3 68a3x2+7a1x +

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·a, x=6a±a√√ 37, x=±a√/10—3a.

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ON THE NATURE AND PROPERTIES OF CURVES, AND THE CONSTRUCTION OF EQUATIONS.

SECTION I.

Nature and Propertics of Curves.

DEF. 1. A curve is a line whose several parts proceed in different directions, and are successively posited towards dif ferent points in space, which also may be cut by one right line in two or more points.

If all the points in the curve may be included in one plane, the curve is called a plane curve; but if they cannot all be comprised in one plane, then is the curve one of double cur

vature.

Since the word direction implies straight lines, and in strictness no part of a curve is a right line, some geometers prefer defining curves otherwise: thus, in a straight line, to be called the line of the abscissas, from a certain point let a line arbi. trarily taken be called the abscissa, and denoted (commonly) by at the several points corresponding to the different values of x, let straight lines be continually drawn, making a certain angle with the line of the abscissas: these straight lines being regulated in length according to a certain law or equation, are called ordinates; and the line or figure in which their extremities are continually found is, in general, a curve line. This definition, however, is not free from objection; for a right line may be denoted by an equation between its abscissas and ordinates, such as y=ax+b.

Curves are distinguished into algebraical or geometrical, and transcendental or mechanical.

Def. 2. Algebraical or geometrical curves, are those in which the relations of the abscissas to the ordinates can be denoted by a common algebraical expression: such, for example, as the equations to the conic sections, given at page 536, &c. vol. i.

Def. 3. Transcendental or mechanical curves, are such as cannot be so defined or expressed by a pure algebraical equation; or when they are expressed by an equation, having one

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