The logarithmic computation is subjoined. Arith. com. log 1323 1. sum 10= 6.8784402 = 0·0858052 0.0429026 const. log. 9.9570974 2.6170003 log Log sin a = 9.5891206 logo- 1.6320232 log ·4285714 = Log sin (60o --▲) const. log. 9.7810061 =0·0429026 = 3 sum - 10 = log x = - · 1·8239087 = log .6666666 =log. Log sin (60°+1)= 9·9966060 const. log. .= 0.0429026 3. sum 10= log - x = 0·0395086=log 1.095238-log. So that the three roots are,, and 23 2; 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 r4 the square roots of these are h = }, √ q = 2 or 2, √ r = Hence, as the value of 16 is negative, the four roots are Ex. 4. Produce a quadratic equation whose roots shall be and . Ex. 5. Produce a cubic equation whose roots shall be 2, 5, Ex. 6. Produce a biquadratic which shall have for the roots 1, 4, 5, and 6 respectively. Ans. x46x3 —21x2 + 146x-120 = 0. Ex. 7. Find x, when x2 + 347x = 22110. 481860, to find x. Ans. x equation 3 20, x = 24093. Sec-1=0. Ex. 11. Find the roots of the Ans. the roots are sin 70°, sin 50°, and 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 23 -x 6 = 0. Ans. 3 x sec 54° 44′ 20′′. Ex. 13. Find the real root of 25x3 +75x 46 0. Ans. 2 cot 74° 27′ 48′′. Ex. 14. Given r4 — 8x3 — 12x2 + 84x-63 = by quadratics. Ex. 15. Given x4 + 36x3 find x, by quadratics. Ex. 16. Given C + 24x3-114x2 -24x+1· O to find r. Ans.✔ 197 - 14, x = 2 ± √5. Ex. 17. Find x, when x-12x-5= 0. Ans. x=1 ± √2, x = when x4 Ex. 18. Find 1 ± 2-1. -12x3 + 47x2 - 72x + 36 = 0. Ans. 1, or 2, or 3, or 6. Ex. 19. Given x5 -5ax1 -80a3x3~68a3x2 + 7a*x+a3 = 0, to find x. Ans. x = ➡ a, x = 6a ± ay/37, x = ± a √10—3a. CHAPTER CHAPTER IX. ON THE NATURE AND PROPERTIES OF CURVES, AND THE CONSTRUCTION OF FQUATIONS. SECTION I. Nature and Properties of Curves. DEF. 1. A curve is a line whose several parts proceed in different directions, and are successively posited towards different 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 arbitrarily 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 in page 532 &c, of vol. 2. 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 VOL. II. Nn of of its terms a variable quantity, or a curve line. Thus, y = log x, y A. şin x, y = A cos x, y = A*, are equations to transcendental curves; and the latter in particular is an equation to an exponential curve Def. 4. Curves that turn round a fixed point or centre, gradually receding from it, are called spiral or radial curves. Def. 5. Family or tribe of curves, is an assemblage of several curves of different kinds, all defined by the same equation of an indeterminate degree; but differently, according to the diversity of their kind. For example, suppose an equation of an indeterminate degree, am-1x=ym: if m ==2, then will ax y2; if m 3, then will a2x y3; ifm = 4, then is a3x = y; &c: all which curves are said to be of the same family or tribe. Def. 6. The axis of a figure is a right line passing through the centre of a curve, when it has one: if it bisects the ordinates, it is called a diameter. Def. 7. An asymptote is a right line which continually approaches towards a curve, but never can touch it, unless the curve could be extended to an infinite distance. Def. 8. An abscissa and an ordinate, whether right or oblique, are, when spoken of together, frequently termed coordinates. ART. 1. The most convenient mode of classing algebraical curves, is according to the orders or dimensions of the equations which express the relation between the co-ordinates. For then the equation for the same curve, remaining always of the same order so long as each of the assumed systems of co-ordinates is supposed to retain constantly the same inclination of ordinate to abscissa, while referred to different points of the curve, however the axis and the origin of the abscissas, or even the inclination of the co-ordinates in different systems, may vary; the same curve will never be ranked under different orders, according to this method. If therefore we take, for a distinctive character, the number of dimensions which the co-ordinates, whether rectangular or oblique, form in the equation, we shall not disturb the order of the classes, by changing the axis and the origin of the abscissas, or by varying the inclination of the co-ordinates. 2. As algebraists call orders of different kinds of equations, those which constitute the greater or less number of dimensions, they distinguish by the same name the different kinds of resulting lines. Consequently the general equation of the first order being 0 = a + ßx+yy; we may refer to the first order all the lines which, by taking x and y for the coordinates, whether rectangular or oblique, give rise to this equation. equation. But this equation comprises the right line alone, which is the most simple of all lines; and since, for this reason, the name of curve does not properly apply to the first order, we do not usually distinguish the different orders by the name of curve lines, but simply by the generic term of lines: hence the first order of lines does not comprehend any curves, but solely the right line. As for the rest, it is indifferent whether the co-ordinates are perpendicular or not; for if the ordinates make with the axis an angle whose sine is μ and cosine, we can refer the equation to that of the rectangular co-ordinates, by making น y = → μ จน and x ==+; which will give for an equation between the perpendiculars and 0 = a + ßt + (~+~)u. μ Thus it follows evidently, that the signification of the equation is not limited by supposing the ordinates to be rightly applied and it will be the same with equations of superior orders, which will not be less general though the co-ordinates are perpendicular. Hence, since the determination of the inclination of the ordinates applied to the axis, takes nothing from the generality of a general equation of any order whatever, we put no restriction on its signification by supposing the co-ordinates rectangular; and the equation will be of the same order whether the co-ordinates be rectangular or oblique. 3. All the lines of the second order will be comprised in the general equation. 0 = a+ ßx+vy +dx2 + ɛxy + (y2 ; that is to say, we may class among lines of the second order all the curve lines which this equation expresses, and y denoting the rectangular co-ordinates. These curve lines are therefore the most simple of all, since there are no curves in the first order of lines; it is for this reason that some writers call them curves of the first order. But the curves included in this equation are better known under the name of CONIC SECTIONS, because they all result from sections of the cone. The different kinds of these lines are the ellipse, the circle, or ellipse with equal axes, the parabola, and the hyperbola; the properties of all which may be deduced with facility from the preceding general equation. Or this equation may be transformed into the subjoined one: --y+ 8x2+6x+x =0: |
