EXAMPLES. Er. 1. Find the roots of the equation x2 + by tables of sines and tangents. 449= 1695 and the equation agrees with the 10= 1.9444827 = 9.1549020 log tan A = 10.6617943 = log tan 77°42′31′′; log tan A = 9.9061115= log tan 38°51'15"}; log 9, as above = 9.5624096 sum 10 = log x = 1·4685211 = log 2941176. This value of x, viz •2941176, is nearly equal to 5. To find 17 whether that is the exact root, take the arithmetical compliment 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 exactly, 5 3.4 and this is manifestly equal to. As to the other root of Ex. 2. Find the roots of the cubic equation 4p3 > 27q2: so that the example falls under the irreducible case. 403 46 = 441 1479 the second term is negative, and The logarithmic computation is subjoined. Arith. com. log 1323 = 6·8784402 half sum = 0·0429026 const. log. Arith. com. const. log =9-9570974 log 414 .. Arith. com. log 403 log sin 3A. = 2.6170003 x= Log sin (60°+A) 3. sum 1.8239087=log 6666666=log. 9.9966060 = 0.0429026 10= log -x=0·0395086= =log1·095238=log}}. 23 So that the three roots are 4, 3, 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 4 — 25x2 + 360, by Euler's rule. 9 2= = p, and z = 4 = 9, and z = =r; 4 25 4 the square roots of these are p = 3, √ q = 2 or 4, √√r=3. Hence, as the value of +b is negative, the four roots are Produce a quadratic equation whose roots shall be Ans. x + 3 = 0. Produce a cubic equation whose roots shall be 2, 11x+30= 0. Ans. x3. 4x2 Er. 6. roots 1, 4, Ex. 7. Produce a biquadratic which shall have for the 5, and 6 respectively. Ans. x+ 6x3 21x146x 120 0. Find x, when x2 + 347x = 22110. Ex. 8. Ans. x 55, x= - 402. Find the roots of the quadratic 2 55 325 = sin 50°, and - Ex. 11. Find the roots of the equation x3- 3.x — 1=0. Ans. the roots are sin 70°, sin 10°, to a radius 2; or the roots are twice the sines of those arcs as given in the tables. Ex. 13. Find the real root of 25x3+ 75x Ans. 2 cot 74° 27′ 48′′. 12x2 + 84x 63 = 0, to Ans. x2 + √7± √11+√7. Ex. 15. Given xa +36x3 — 400x2 — 3168x + 7744≈0, to find x, by quadratics. Ans. 11+√209. Given 4+24x3- 114x2 - 24x+10, to Er. 19. Given xs-5ax4-80a2x3-68a3x2+7a*x+a3=0, to find x. Ans. xa, x = 6a ±a√√37, x = ± a√/10-3a. CHAPTER CHAPTER IX. ON THE NATURE AND PROPERTIES OF CURVES, AND THE CONSTRUCTION OF EQUATIONS. 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 comprized 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 r, 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 the first chapter of this volume. Def. 3. Transcendental or mechanical curves, are such as cannot be so defined or expressed by a pure algebraical equasion; or when they are expressed by an equation, having one of of its terms a variable quantity, or a curve line. Thus, y = log x, y A. sin 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, a"-1x =y": if m=2, then will ax = y; if m = 3, then will a'x = y3; if m = 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 = a + B x + ry; we may refer to the L first |
