20. Cones are drawn, touching an ellipsoid, from any two points of a similar, similarly situated, and concentric ellipsoid. Show that they intersect in two plane curves. Find the locus of the vertices of the cones when these plane sections are at right angles to one another. 21. Find the locus of the points of contact of tangent planes which are equidistant from the centre of a surface of the second order. 22. From a fixed point 4, on the surface of a given sphere, draw any chord AD; let D' be the second point of intersection of the sphere with the secant BD drawn from any point B; and take a radius vector AE, equal in length to BD, and in direction either coincident with, or opposite to, the chord AD: the locus of E is an ellipsoid, whose centre is A, and which passes through B. (Hamilton, Elements, p. 227.) 23. Show that the equation. - l2 (e2 — 1) (e + Saa') = (Sap)2 — 2e Sap Sa′p + (Sa′p)2 + (1—e2)p2, where e is a variable (scalar) parameter, represents a system of confocal surfaces. (Ibid. p. 644.) 24. Show that the locus of the diameters of which are parallel to the chords bisected by the tangent planes to the cone is the cone Spуp = 0 S.ppy1pp = 0. 25. Find the equation of a cone, whose vertex is one summit of a given tetrahedron, and which passes through the circle circumscribing the opposite side. 26. Show that the locus of points on the surface Spop = 1, the normals at which meet that drawn at the point p = w, is on the cone S.(p-w) pwpp = 0. 27. Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane. 28. Show that the locus of the pole of the plane 29. Show that the equation of the surface generated by lines drawn through the origin parallel to the normals to 30. Common tangent planes are drawn to 2Sλp Sup+(p-Sλu) p2 1, and Tp = find the value of h that the lines of contact with the former 32. Tangent cones are drawn from every point of S(p-a)p(p-a) = n2, to the similar and similarly situated surface Spopp = 1, show that their planes of contact envelop the surface (Sapp-1)2= n2 Spøp. 33. Find the envelop of planes which touch the parabolas p = at2 +ßt, p = ar2+yτ, where a, ß, y form a rectangular system, and t and τ are scalars. 34. Find the equation of the surface on which lie the lines of contact of tangent cones drawn from a fixed point to a series of similar, similarly situated, and concentric ellipsoids. 35. Discuss the surfaces whose equations are and Sap SBP = Syp, S'ap+S.aßp = 1. 36. Show that the locus of the vertices of the right cones which touch an ellipsoid is a hyperbola. CHAPTER IX. GEOMETRY OF CURVES AND SURFACES. 279. We have already seen (§ 31 (7)) that the equations where a represents one of a set of given vectors, and ƒ a scalar function of scalars t and u, represent respectively a curve and a surface. We commence the present brief Chapter with a few of the immediate deductions from these forms of expression. We shall then give a number of examples, with little attempt at systematic development or even arrangement. 280. What may be denoted by t and u in these equations is, of course, quite immaterial: but in the case of curves, considered geometrically, t is most conveniently taken as the length, 8, of the curve, measured from some fixed point. In the Kinematical investigations of the next Chapter t may, with great convenience, be employed to denote time. 281. Thus we may write the equation of any curve in space as where is a vector function of the length, s, of the curve. Of course it is only a linear function when the equation (as in § 31 ()) represents a straight line. 282. We have also seen (§§ 38, 39) that is a vector of unit length in the direction of the tangent at the extremity of p. At the proximate point, denoted by s+ds, this unit tangent vector becomes But, because we have p's+p's ds + &c. To's = 1, S.p's p's = 0. Hence "s is a vector in the osculating plane of the curve, and perpendicular to the tangent. Also, if de be the angle between the successive tangents o ́s and p's+p's ds + so that the tensor of curvature at the point s. we have "s is the reciprocal of the radius of absolute 283. Thus, if OP = ps be the vector of any point P of the curve, and if C be the centre of curvature at P, we have is the equation of the locus of the centre of curvature. is the vector perpendicular to the osculating plane; and is the tortuosity of the given curve, or the rate of rotation of its osculating plane per unit of length. 284. As an example of the use of these expressions let us find the curve whose curvature and tortuosity are both constant. We have curvature= To's = Tp"'= c. |