: 395. PROP. 3. If the parabola (z=-4 m'x) move parallel to itself, so that its vertex describe the parabola (y=4m'x), it will generate the hyperbolic paraboloid. The equation to the generating parabola is z2= -4m'x...(1), and that to the directrix Y= 4mX...(2). In any position the equations to the generating curve will be y=ß, z2 = − 4 m'x+a..... (3). And, since it must pass through a point (x = X, y=Y, z=0,) of the directrix, we have in equation (3), therefore, equation (2) becomes, by substitution, CHAP. IX. ON THE SECTIONS OF SURFACES OF THE SECOND ORDER 396. PROP. 1. To find the nature of the curve formed by the intersection of a surface and a plane. I. Let the surface be supposed to have a centre. Then, the general equation is, (353) Ax2 + By2+ Cz2 + D=0. In order to determine the section made by a plane, we must x=x' cos + y' cos 0 cos 0, y=x' sin -y' cos e cos 0, z=y' sin 0. The result, developed and arranged, will be (A cos2 +B sin3) x22 + {C sin2 0+cos2 0 (A sin2+B cos3p)}y” +2 (AB) sin cos o cos 0.x'y' + D=0....(e). Wherefore, the curve of intersection is a line of the second order. The species of the curve will depend, (Art. 90 and 201), on the sign of the quantity 4(A-B2) sin2 cos cos 0-4 (A cos2 + B sin2p) { Csin' 0+cos (A sin'+Bcos)}, or, when the expression is reduced, of the quantity -AB cos20-BC sino 0 sin2 - AC sin2 0 cos2 ...... (1). Ꭱ Ꭱ Let us now suppose the surface to be (1) An ellipsoid. Then, the coefficients A, B, C being all positive, the above quantity is essentially negative. Consequently, the section of an ellipsoid, by a plane, must be an ellipse, or one of its varieties. (2) An hyperboloid of one, or of two sheets. Then, since any two of the coefficients A, B, C may be positive or negative, and the remaining one negative or positive, the above quantity (1), may be positive, or negative, or=0. Consequently, the section of either species of hyperboloid by a plane, may be an ellipse, hyperbola, or parabola. We have already arrived at the same conclusion, (Art. 344, and 345), in the case of the cone, which, in fact, is only a variety of either kind of hyperboloid. II. Let the surface be supposed not to have a centre. Then, the general equation is of the form My2+M'z2+Px=0. Following the same steps as in the first part of the proposition, we obtain, for the equation to the curve of intersection, 2 (Mcos e cos'+M'sin'0) y2+ M'sin'p.x2-2M sino coscoso.xy + P cos o cos 0.y + P cos . x=0................ (e'). The curve is, therefore, a line of the second order, whose species depends on the sign of the quantity -MM' sin sin2 0. (1) Let the surface be the elliptic paraboloid. Then because M, M' have the same sign, the above quantity is essentially negative. Hence, the section is an ellipse. The ON SURFACES OF THE SECOND ORDER. quantity-MM' sin2 sin3 will = 0, if either 0 or =0. The sections, therefore, in either case, will be parabolas, (208). Consequently, the sections of the elliptic paraboloid, by a plane, may be an ellipse or parabola, or a variety of either curve. (2) Let the surface be the hyperbolic paraboloid. Then, M, M' having different signs, the quantity is necessarily positive. Whence, the section is au hyperbola. If, however, or 0=0, the section will, in either case, be a parabola. Consequently, the section of the hyperbolic paraboloid, may be an hyperbola or a parabola, or a variety of either curve. 397. COR. It appears, from the second case of the Proposition, that the section of the elliptic paraboloid can never be an hyperbola, nor the section of the hyperbolic paraboloid, an ellipse. 398. PROP. 2. To investigate the conditions under which the section of a surface of the second order, may be a circle. I. Let the surface have a centre. Then, in order that the section may be a circle, the coefficients of y' and x2, in the equation to the section, must be equal, and the term involving ry, must vanish, (204). Whence, referring back to equation (e), Art. 396, A cos2 +Bsin2 = C sin3 0+ cos2 0 (A sin2 +B cos34) ..... (m) Now the latter of these equations will be satisfied, if (n) |