In the figure, м'ст' represents what is termed the superior conchoid, and GBMDMBm the inferior conchoid. The point B is called the pole of the conchoid; and the curve may be readily constructed by radial lines from this point, by means of the polar equation z = cos. ±a. It will merely be re quisite to set off from any assumed point a, the distance AB = b; then to draw through в а right line MLM making any angle & with CB, and from L, the point, where this line cuts the directrix AY (drawn perpendicular to CB) set off upon it LM' = LM = a; so shall m' and m be points in the superior and inferior conchoids respectively. Ex. 4. Let the principal properties of the curve whose equation is yr" = a"+', be sought; when n is an odd number, and when n is an even number. Ex. 5. Describe the line which is defined by the equation xy + ay + cy = bc + bx. Er. 6. Let the Cardioide, whose equation is y-6ay3 + (2.r2 + 12a2) y2 - (6ax2 + 8a3)y + (x2 + 3a2) x2 = 0, be proposed. Ex. 7. Let the Trident, whose equation is xy = ax3 + bx2 + cx + d, be proposed. Er. 8. Ascertain whether the Cissoid and the Witch, whose equations are found in the preceding problem, have asymptotes. PROBLEM III. To determine the Equation to any proposed Curve Surface. Here the required equation must be deduced from the law or manner of construction of the proposed surface, the reference being to three co-ordinates, commonly rectangular ones, the variable quantities being x, y, and z. Of these, two, namely x and y, will be found in one plane, and the third z will always mark the distance from that plane. Ex.1. Let the proposed surface be that of a sphere, FNG. The position of the fixed point a, which is the origin of the co-ordinates AP, PM, MN, being arbitrary; let it be supposed, for the greater convenience, that it is at the centre of the sphere. Let MA, NA, be drawn, of which the latter is manifestly equal to the radius F V T N G M A P Z of the sphere, and may be denoted by r. Then, if AP = x, PM = Y, MN = %; the right-angled triangle APM will give AM2 AM2 = AP2 + PM2 = r2 + y2. In like manner, the rightangled triangle AMN, posited in a plane perpendicular to the former, will give AN2 = AM2 + MN2, that is, r2 = x2+y2+z2; or z2 = r2 - x2 - y2, the equation to the spherical surface, as required. Scholium. Curve surfaces, as well as plane curves, are arranged in orders according to the dimensions of the equations, by which they are represented. And, in order to determine the properties of curve surfaces, processes must be employed, similar to those adopted when investigating the properties of plane curves. Thus, in like manner as in the theory of curve lines, the supposition that the ordinate y is equal to 0, gives the point or points where the curve cuts its axis; so, with regard to curve surfaces, the supposition of z = 0, will give the equation of the curve made by the intersection of the surface and its base, or the plane of the coordinates x, y. Hence, in the equation to the spherical surface, when z = 0, we have r2 + y2 = r2, which is that of a circle whose radius is equal to that of the sphere. See p. 31. Ex. 2. Let the curve surface proposed be that produced by a parabola turning about its axis. Here the abscissas x being reckoned from the vertex or summit of the axis, and on a plane passing through that axis; the two other co-ordinates being, as before, y and z; and the parameter of the generating parabola being p: the equation of the parabolic surface will be found to be z2 + y2 px = 0. Now, in this equation, if z be supposed = 0, we shall have y2 = px, which (pa. 31) is the equation to the generating parabola, as it ought to be. If we wished to know what would be the curve resulting from a section parallel to that which coincides with the axis, and at the distance a from it, we must put z = a; this would give y2 = px - a2, which is still an equation to a parabola, but in which the origin of the abscissas is distant from the vertex before assumed by the quantity 42 P Ex. 3. Suppose the curve surface of a right cone were proposed. Here we may most conveniently refer the equation of the surface to the plane of the circular base of the cone. In this case, the perpendicular distance of any point in the surface from the base, will be to the axis of the cone, as the distance of the foot of that perpendicular from the circumference VOL. III. P 1 (measured In conc B is read of t qui AB an cu it ar on 210 mesured on a radius), to the radius of the base: that is, if the ves of be estimated from the centre of the base, and be the radius, & will vary as r√(x2 the simplest equation =-(+y2), or r + y2). Conse of the conic surface, will be 2iz + z2 = x2 + y2. New trem this the nature of curves formed by planes cut ting the cone in diferent directions, Leti, be supposed, first, that the the the may readily belanes cutcutting plane is inchferred. angle of 45, and passes units centre: then willz, and this value of the plane is manifestly an SO sub 2r.x=y, which is the equation of the projection of the curve sticute for it in the equation of the surface, will give r2 of the cone's base: and this (art. 3 of this chap.) equation to a parabola. Or, taking the thing more generally, let it be supposed that the cutting plane is the hat of 1 to m: then will mx = z, and m2x2 = 22. These bes ituted for z and z2 in the equation of the surface, will give, for the equation of the projection of the section on the 2mx + (m2 - 1)x2 = y2. Now this plane of the base, 12 be greater than unity, or if the cutting plane equation, if m between the vertex of the cone and the parabolic sec situated, that the ratio of a to z shall pass tion, will be that of an hyperbola: and if, on the contrary, the cutting plane pass between the parabola and the base, i. e. if m be less than unity, the term (m2-1)x" will be negative, when the equation will obviously designate an ellipse. Schol. It might here be demonstrated, in a nearly similar manner, that every surface formed by the rotation of any conic section on one of its axes, being cut by any plane whatever, will always give a conic section. For the equation of such surface will not contain any power of r, y, or z, greater than the second; and therefore the substitution of any values of z in terms of x or of y, will never produce any powers of x or of y exceeding the square. The section therefore must be a line of the second order. See, on this subject, Hutton's Mensuratión, part iii, sect. 4. Er. 3. Let the equation to the curve surface be xyz = a3. Then will the curve surface bear the same relation to the solid right angle, which the curve line whose, equation is ry = a2 bears to the plane right angle. That is, the curve surface will be posited between the three rectangular faces bounding such solid right angle, in the same manner as the equilateral hyperbola is posited between its rectangular asymptotes. And in like manner as there may be 4 equal equilateral teral hyperbolas comprehended between the same rectangular asymptotes, when produced both ways from the angular point; so there may be 6 equal hyperboloids posited within the 6 solid right angles which meet at the same summit, and all placed between the same three asymptotic planes. SECTION II. On the Construction of Equations. PROBLEM I. To Construct Simple Equations, Geometrically. : HERE the sole art consists in resolving the fractions, to which the unknown quantity is equal, into proportional terms; and then constructing the respective proportions, by means of probs. 8, 9, 10, and 27 Geometry. A few simple examples will render the method obvious. 1. Let x; then c:a :: bx. Whence x may be found by constructing according to prob. 9 Geometry. ab abc 2. Let x = First construct the proportion d: a::b: de gc , which 4th term call g; then =; ore:c::g: x. 3. Let x = a2-2.2 C e Then, since a2-b2= (a+b)x(a−b); it will merely be necessary to construct the proportion c: a + b :: a-b:x. hc first case i. So shall x = g - i, the difference of those a lines, found by construction. First find, the fourth propor tional to b, a and f, which make = h. Then x = or, by construction it will be h+c:a-d::a : x. a2+22 a(u-d) h+c; 6. Let x = Make the right-angled triangle ABC such AC2 that the leg AB = a, BC = b; then AC = √(AB2 + BC2) = √(a2 + b2), by th. 34 Geom. Hence Construct therefore the proportion C:AC::AC:x, and the unknown quantity will be found, as required. x= C a2 + cd 2 C B mean proportional between AC = c, and 7. Let x = h+c CB = d, that is, find CD = √ cd. Then make CE = a, and join De, which will evidently be = √(a2 +cd). Next on AC A EB any line EG set off EF = h + c, EG = ED; and draw GH parallel to FD, to meet DE (produced if need be) in H. So shall EH be = x, the third proportional to h + c, and √(a2 + cd), as required. Note. Other methods suitable to different cases which may arise are left to the student's invention. And in all constructions the accuracy of the results, will increase with the size of the diagrams; within convenient limits for operation. PROBLEM II. To Find the Roots of Quadratic Equations by Construction. In most of the methods commonly given for the construction of quadratics, it is required to set off the square root of the last term; an operation which can only be performed accurately when that term is a rational square. We shall here describe a method which, at the same time that it is very simple in practice, has the advantage of showing clearly E G A C B D H F the relations of the roots, and of dividing the third term into two factors, one of which at least may be a whole number. In order to this construction, all quadratics may be classed 1. One general mode of construction will include the first two of these forms. Let x2幹ax - bc = 0, and b be greater than c. Describe any circle ABD having its diameter not less than the given quantities a and b - c, and within this circle inscribe |
