EQUATIONS TO CURVE SURFACES. 7. Then, if AP='x, PM = y, MN; the right-angled triangle APM will give AM2 = AP2 + PM2 = x2 + y'. In like manner, the right-angled triangle AMN, posited in a plane perpendicular to the former, will give ANAM2+ MN, that is, j2 = x2 + y2 + z2; or z2 = r23 — x2-y, 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 equa tions, 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 Thus, in like manner as in the properties of plane curves. 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 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 sur= r2, which is that of a face, when z = 0, we have x2 + y2 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. z= 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 equa tion 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 y=pr, which (p. 538, vol. i.) is the equation to the generat. If we wished to know what ing parabola, as it ought to be. would be the curve resulting from a section parallel to that which coincides with the axis, and at the distance a from it, - a2, which is we must put za; this would give y3 = pr still an equation to a parabola, but in which the origin of the abscissas is distant from the vertex before assumed by the a3. quantity Р Ex. 3. Suppose the curve surface of a right cone were proposed. Here we may most conveniently refer the equation of the In this surface to the plane of the circular base of the cone. 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 (measured on a radius), to the radius of the base; that is, if the values of x be estimated from the centre of the base, and ☛ be the radius, z will vary as r - √ (x2 + y2). Consequently, the simplest equation of the conic surface, will be 2 ~ r = = √(x2 + y2), or r2 — 2rz +z2 = x2 + y2. Now, from this, the nature of curves formed by planes cut. ting the cone in different directions, may readily be inferred. Let it be supposed, first, that the cutting plane is inclined to the base of a right-angled cone in the angle of 45°, and passes through its centre: then will z = x, and this value of z substituted for it in the equation of the surface, will give r2 — 2rx= y, which is the equation of the projection of the curve on the plane of the cone's base and this (art. 3 of this chap.) is manifestly an equation to a parabola. - - Or, taking the thing more generally, let it be supposed that the cutting plane is so situated, that the ratio of x to z shall be that of 1 to m: then will mx = z, and m2r2 = z2. These substituted for z and z2 in the equation of the surface, will give, for the equation of the projection of the section on the plane of the base, ra 2mx+ (m2-1) x2 = y2. Now this equation, if m be greater than unity, or if the cutting plane pass between the vertex of the cone and the parabolic sec. 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) x2 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 x, y, or z, greater than the second; and therefore the substitution of any values of z in terms of r or of y, will never produce any powers of or of y exceeding the square. The section therefore must be a line of the second order. See, on this subject, Hutton's Mensuration, part iii. sect. 4. Ex. 4. 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 xy= a 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 equila. teral hyperbolas comprehended beween 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 = ab then cab: x. Whence may be found by constructing according to prob. 9 Geometry. First construct the proportion d:a::b: which 4th term call a2 ba 3. Let x = a2=63. Then, since a2—b2=(a+b)X(a—b) ; C it will merely be necessary to construct the proportion c : a+ba b: x. 5. Let x = lines, found by construction. a2b-bad or, by construction it will be h + c:a — First find, the fourth propor that the leg AB = a, BC= b; then AC = √ (Ab2 + BC3) = √ (a2 + b2), by th. 34 Geom. Hence Construct therefore the proportion x= AC2 C C: AC AC: 2, and the unknown quantity will be found as required. a2 + cd AC B G E B 7. Let x = First, find CD a mean proportional between AC = c, and CB = d, that is, find CD ✔ cd. Then make (E = a, and join DE, which will evidently be√(a2 + cd). Next on any line EG set off EF = = h +c, EG = ED; and draw GI parallel to FD, to meet DE (produced if need be) in H. So shall EH ber, 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 ope ration. PROBLEM 11. To find the roots of quadratic equations by construction. B E G C. 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 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 under 4 forms: viz. x2 + ax bc 0. 1. 2. - ax bc = 0. 3. x2 + ax + bc = 0. 4. x2 - ax + bc = 0. H 1. One general mode of construction will include the first two of these forms. Let x ax bc0, and b be greater than c. Describe any circle ABD having its diameter not less c, and within this circle c, both from any b than the given quantities a and b inscribe two chords, AB = α, AD = common assumed point a. Then, produce AD to F so that Dr=c, and about the centre c of the former circle, with the radius CF, describe another circle, cutting the chords ad, ab, produced, in F, E, G, H: so shall AG be the affirmative and AH the negative root of the equation a+ax-bc = 0; and contrariwise AG will be the negative and AH the affirmative root of the equation x2 -ax bc = 0. For, AF or AD + df = b, and DF or AE=c; and, making AG or BH = x, we shall have AH = a + x; and by the property of the circle EGFH (theor. 61 Geom.) the rectangle EA. AF = GA. AH, or be (a + x)x, or again by transposition x2+ax — - bc =0. Also if AH be= = , we shall have AG or BH or AH - - AB — — x — a: and conseq. GA . AH = x2 + ax, as before. So that, whether AG be = 2, or AH = — x, we shall always have r2 + ax - bc = 0. And by an exactly similar process it may be proved that AG is the negative, and AH the positive root of x2-ax bc = 0. Cor. In quadratics of the form x2 + ax-bc = 0, the positive root is always less than the negative root; and in those of the form x-ax-bc=0, the positive root is always greater than the negative one. B H GAE 2. The third and fourth cases also are comprehended under one method of construction, with two concentric circles. Let x2 + ax + bc = 0. Here describe any circle ABD, whose diameter is not less than either of the given quantities a and b+c; and within that circle inscribe two chords AB, AD = b+c, both from the same point a. Then in AD assume DF = c, and about c the centre of the circle ABD, with the radius cr describe a circle, cutting the chords AD, AB, in the points F, E, G, HI: so shall ag, ah, be the two positive roots of the equation x2 ax + bc and the two negative roots of the equation x2+ar + be = 0. The demonstration of this also is similar to that of the first case. 0, Cor. 1. If the circle whose radius is CF just touches the chord AB, the quadratic will have two equal roots; which can only happen when a2 = bc. Cor. 2. If that circle neither cut nor touch the chord AB, the roots of the equation wil! be imaginary; and this will |