sum Arith. com. log. 7 süm + 10 = log 88= 10= log tan a = 10·6617943 = log tan 77°42′31′′2 ; sum 5 10 = log x = 1·4685211 = log ·2941176. 1 3.4exactly, and 441 x + this is manifestly equal to 1 As to the other root of the 339 12716 17. 748 equation, it is equal to 403 46 X3 O, by a table of sines. 403 147 3.2291697 5.8956495 19.1248192 9.5624096 5 and the equation agrees with the 1695 and x=tan A=√12716" 12716' 1695 12716* 46 Here p the second term is negative, and 441 1472 1 403 2/3·441 ✓1323 1695 5 The three values of a therefore, are 1612 sin A√1323 x= sin 2. sum - 10 1612 1323 1612 9.7810061 0.0429026 1-6320232-log-4285714=log. 9.9966060 0.0429026 271 log sin 68°32′18′′. 1.8239087=log·6666666=log. 3. sum 10= log -x=0·0395086-log1.095238-log23. 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. 9 25 2 ===p, and z=4=q, and z = Ex. 3. Find the roots of the biquadratic x1-25x2 -+60x36=0, by Euler's Rule. Here a=25, b=-60, and c=36; therefore 25 625 769 16. ƒ = = 8 = +9= and h 16 Consequently the cubic equation will be 25 23 723 769 225 =0. The three roots of which are 225 g; 4. the square roots of these are p=2; √9=2 or 4, √r. Hence, as the value of 16 is negative, the four roots are 1st. x = 3+4-2=1, 4th. x=-4-4=-6. Ex. 4. 272 SOLUTIONS OF EQUATIONS. Ex. 4. Produce a quadratic equation whose roots shall be and 4. Ans. x2-x+3=0. Ex. 5. Produce a cubic equation whose roots shall be, 2, 5, and -3. Ans. x3- 4x3-11x+30=0. Ex. 6. Produce a biquadratic which shall have for the roots 1, 4, 5, and 6 respectively. { Ans. x-6x3-21x2+146x-120-0. Ex. 7. Find x, when x2+347x=22110. Ex. 8. Find the roots of the quadratic x2. Ex. 9. Solve the equation x2 8x3 Ans. x=55, x= X 12 Ans. x 10, x 695 Ans. x=5, x= 264 25 X Ex. 10. Given x2-24113x-481860, to find x. Ans. x 20 x=24093. Ex. 11. Find the roots of the equation x3-3x-1=0. Ans. the roots are sin 70°, sin 50o, and sin 10o, 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 3× sec 54° 44′ 20′′. Ex. 14. Given x4 Ans. x=1±√2, x—— 1±2√−1. 12x3+47x2 —72x+36=0. Ex. 18. Find x, when x4 402. 325 6 65 12 139 25. Ans. x=1, or 2, or 3, or 6. 80a3x3 — 68a2x2+7α1x+5=0, 4 Ans. ≈ —α, x=6a±α √37, x=±a√ 10-3a. ON [ 273 ] ON THE NATURE AND PROPERTIES OF CURVES, SECTION Í. 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 x 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 equa tion, 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. 36 of 274 NATURE AND PROPERTIES OF CURVES. of its terms a variable quantity, or a curve line. Thus, y the Def. 4. Curves that turn round a fixed point or centre, gradually receding from it, are called spiral or radial curves. .1 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=y; if m=3, then will a2x=y3; if m=4, then is a3x=y*, &c. : all which curves are said to be of the same family or tribe. ןי J 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 dif ferent 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 + B x + ry; 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, ܪ܂ |