In this simple case, the equation to the surface is easily found by the method in article (468). Taking the figure in that article, and supposing every section, like P Q, to be an ellipse, whose axes a, and y, are always proportional to the axes a and b of an ellipse whose centre is in A Z, and at a distance c from A, we have the equation to P Q 515. Let the directrix be a parabola parallel to x y, and vertex in the The equations to the directrix and generatrix are axis of z. Y2 = p X at the points of junction we have x − a = α (z 516. Let the vertex or centre of the cone be at the origin.. abco, and the equation to a cone whose directrix is {y2 = p x, x = d } and whose vertex is at the origin, is dy' = pxz. 517. The following method of finding the equation to a right cone whose vertex is at the origin, is sometimes useful. Let the length of the axis of the cone be k, and suppose this axis to pass through the origin, and be perpendicular to a given plane or base whose equation therefore will be of the form a x + By + y z = k where a, B, y are the co-sines of the angles which k makes with the axis of x, y, and z (410). Also suppose x, y, and z to be the co-ordinates of a point on the circumference of this base, and let O be the angle which the generatrix of the cone makes with its axis, then by the property of the right-angle triangle we have the equation k = if (x2 + y2 + zo) cos. 0 Hence by equating the values of k we have the equation, (∞ x + By + y z)3 = (x2 + y2 + ~2) (cos. 0)3. And this is the equation to any point in the surface, since a, B, y remain the same for a plane parallel to the base and passing through any point (ry z) of the surface. If the axis of the cone coincides with the axis of 2, we have a 80 and y = 1; ~2 = (x2 + y2 + *°) (cos. 0)2 518. To find the curve of intersection of a plane and an oblique cone, we may suppose the cutting plane to pass through the origin of co-ordinates without detracting from the generality of the result. Substituting for x, y, z, in the equation, their values in 455, we readily find that the sections are lines of the second order and their varieties. ON CONOIDAL SURFACES. 519. Definition. A conoidal surface is generated by the movement of a straight line constantly parallel to a plane, one extremity of the line moving along a given straight line, the other describing a given curve. We shall commence with a simple case. Let the axis of z be one directrix, and let the generatrix be parallel to the plane of ry: then the equations to the generatrix in any one position are y = ax Now it is evident that when a point moves on the surface without quitting the generatrix, a and b are both constant, but when it passes from one position of the generatrix to another a and b are both variable; hence these quantities, being constant together and variable together, are functions of one another. .. b (a) or substituting their values. 2=9 x which is the general equation to all conoidal surfaces. 520. The form of the function will depend upon the nature of the second directrix. By combining the equations to the generatrix and this directrix, we may, as before, eliminate x, y, z, and arrive at an equation between 6 and a, we must then substitute the values of b and a, their general values z and and we shall obtain the equation to the particular conoidal surface. y Ꮖ 521. Let the second directrix be a circle parallel to yz, and the centre in the axis of x, therefore the equations to this directrix are This surface partaking of the form and generation of both the cone and the wedge, was called the cono-cuneus by Wallis, who investigated many of its properties. If the axis of a be one directrix, and the other be a circle parallel to x 2, and the generatrix be parallel to y z, the equation is 522. Let the axis of z be one directrix, any straight line the other, and let the generatrix move parallel to x y. Then the equations to the second directrix are X = μZ+m Y =ν Ζ + η Also the equations to the generatrix being y = ax, zb, we have at the points of junction Z=2=b Y = y = vb + n 523. Let the axis of z be one directrix, and let the second directrix be the thread of a screw whose axis is coincident with the axis of z. The thread of a screw, or the curve called the helix, is formed by a thread wrapped round the surface of a right cylinder, so as always to make the same angle with the axis; or if the base of a right-angled triangle coincide with the base of the cylinder, and the triangle be wrapped round the cylinder, the hypothenuse will form the helix A P. To find the equations to the helix, Let the centre of the cylindrical base be the origin of rectangular axes. CM = x, MQ = y, P Q = z and the radius of the cylinder = a. Then P Q bears a constant ratio to A Q; namely, that of the altitude to the base of the describing triangle . PQe A Q and AQ is a circular arc whose sine is y and radius a: ..ze a sin. -1 Z And these are the equations to the projections of the helix. To return to the problem, which is to find the surface described by a line subject to the conditions that it be parallel to the base of the cylinder, that it passes through the axis, and that it follows the course of the helix. The equations to the directrix (if c be the interval between two threads) are a ze a sin. + c And the equations to the generatrix being y = a x, z = b; we have This surface is the under side of many spiral staircases. 524. A straight line passes through two straight lines whose equations are x = a, y = b; and x = a1, z = b1; and also through a given curve z= f(y) in the plane of zy; to find the equation to the surface traced out by the straight line. Then since this line meets the three given lines, we have the following equations β b = a + p, a1 = a b1 + m, =ƒ(- B+ n) a α α We must now eliminate a, b, m, n from these equations, and that to the generatrix. 525. The following problem is easily solved in the same manner. To find the equation to a surface formed by a straight line moving parallel to the plane of x 2, and having its extremities in two given curves z = f (y) on z y, and x = 4 (y) on x y. 526. In questions of this kind some care is requisite in selecting the position of the axes and co-ordinate planes, so that the equations, both those given and those to be found, may present themselves in the simplest form. For example, to find the surface formed by the motion of a straight line constantly passing through three other given straight lines; Take three lines parallel to the given lines for the axes of co-ordinates; then the equations to the three directrices are and the equations to the generating line in any position are x = x z + a, y: = Bz + b, β and consequently y = rc, where cb α Ba; α Then since this line meets each of three given lines, we have the following equations: β b1 = a1 + c ; α2 = a c2 + a ; b2 = ß c3 + b. α We must now eliminate a, b, a, ß from these three equations and that to the generatrix; by subtraction we have hence, eliminating a and B, we have the required equation (3ɔ - ≈) (q − h) (n − x) = (3ɔ — z) (®q — h) ('v − x) which is of the second order, since the term x y z disappears. Hymers's Anal. Geom. p. 23, Cambridge, 1830. See CHAPTER IX. ON CURVES OF DOUBLE CURVATURE. 527. Definition. A curve of double curvature is one whose generating point is perpetually changing not only the direction of its motion, as in plane curves, but also the plane in which it moves. If a circle be described on a flat sheet of paper, it is a plane curve; let the sheet of paper be rolled into a cylindrical form, then the circle has two curvatures, that which it originally had, and that which it has acquired by the flexion of the paper, hence in this situation it is called a curve of double curvature. 528. Curves of double curvature arise from the intersection of two surfaces; for example, place one foot of a pair of compasses on a cylindrical surface, let the other in revolving constantly touch the surface, it will describe a curve of double curvature, which, though not a circle, has yet all |