Hence op is perpendicular to the tangent-plane at the extremity of p. The equation of this plane is therefore ( being the vector of any point in it) Spp (p) = 0, or, by the equation of the cone, διφρ = 0. 246.] The equation of the cone of normals to the tangent-planes of a given cone can be easily formed from that of the cone itself. For we may write it in the form S(p-1pp) pp = 0, and if we put op=σ, a vector of the new cone, the equation becomes Numerous curious properties of these connected cones, and of the corresponding spherical conics, follow at once from these equations. But we must leave them to the reader. 247.] As a final example, let us find the equation of a cyclic cone when five of its vector-sides are given―i. e. find the cone of the second order whose vertex is the origin, and on whose surface lie the vectors a, B, y, d, e. If we write 0 = = S.V (VaßVòe) V (VßyVep)V (VydVpa), ...... (1) we have the equation of a cone whose vertex is the origin-for the equation is not altered by putting xp for p. Also it is the equation of a cone of the second degree, since p occurs only twice. Moreover the vectors a, ß, y, d, e are sides of the cone, because if any one of them be put for p the equation is satisfied. Thus if we put ẞ for p the equation becomes 0 = S.V (VaẞVde)V (VßyV€ß)V (VyòVßa) = S.V (VaßVde) { VßaS.VyòVߥVeß—VyòS.VßаVßуVeß}. The first term vanishes because S.V (VaẞVde) Vẞa = 0, and the second because S.VẞaVBуVeẞ = 0, since the three vectors Vẞa, Vẞy, Veß, being each at right angles to ẞ, must be in one plane. As is remarked by Hamilton, this is a very simple proof of Pascal's Theorem-for (1) is the condition that the intersections of the planes of a, ẞ and d, e; ß, y and e, p; y, d and p, a; shall lie in one plane; or, making the statement for any plane section of the cone, that the points of intersection of the three pairs of opposite sides, of a hexagon inscribed in a curve, may always lie in one straight line, the curve must be a conic section. EXAMPLES TO CHAPTER VII. 1. On the vector of a point P in the plane Sap =1 a point Q is taken, such that QO.OP is constant; find the equation of the locus of Q. 2. What spheres cut the loci of P and Q in (1) so that both lines of intersection lie on a cone whose vertex is 0? 3. A sphere touches a fixed plane, and cuts a fixed sphere. If the point of contact with the plane be given, the plane of the intersection of the spheres contains a fixed line. Find the locus of the centre of the variable sphere, if the plane of its intersection with the fixed sphere passes through a given point. 4. Find the radii of the spheres which touch, simultaneously, the four given planes Sap = 0, SBp = 0, Syp = 0, Sop=1. [What is the volume of the tetrahedron enclosed by these planes ?] 5. If a moveable line, passing through the origin, make with any number of fixed lines angles 0, 01, 02, &c., such that a cos.0+a, cos.02+.... = constant, where a, a,...... are constant scalars, the line describes a right cone. 6. Determine the conditions that may represent a right cone. Spop=0 7. What property of a cone (or of a spherical conic) is given directly by the following form of its equation, Σιρκρ =0? 8. What are the conditions that the surfaces represented by Sppp 0, and S.pkp = 0, = = may degenerate into pairs of planes ? Σιρκρ 9. Find the locus of the vertices of all right cones which have a common ellipse as base. 10. Two right circular cones have their axes parallel, shew that the orthogonal projection of their curve of intersection on the plane containing their axes is a parabola. 11. Two spheres being given in magnitude and position, every sphere which intersects them in given angles will touch two other fixed spheres and cut a third at right angles. 12. If a sphere be placed on a table, the breadth of the elliptic shadow formed by rays diverging from a fixed point is independent of the position of the sphere. 13. Form the equation of the cylinder which has a given circular section, and a given axis. Find the direction of the normal to the subcontrary section. 14. Given the base of a spherical triangle, and the product of the cosines of the sides, the locus of the vertex is a spherical conic, the poles of whose cyclic arcs are the extremities of the given base. 15. (Hamilton, Bishop Law's Premium Ex., 1858.) (a.) What property of a sphero-conic is most immediately indicated by the equation = 0 a ρ (b.) The equation (Ap)2+(Sup)2: also represents a cone of the second order; A is a focal line, and μ is perpendicular to the director-plane corresponding. (c.) What property of a sphero-conic does the equation most immediately indicate? 16. Shew that the areas of all triangles, bounded by a tangent to a spherical conic and the cyclic arcs, are equal. 17. Shew that the locus of a point, the sum of whose arcual distances from two given points on a sphere is constant, is a spherical conic. 18. If two tangent planes be drawn to a cyclic cone, the four lines in which they intersect the cyclic planes are sides of a right cone. 19. Find the equation of the cone whose sides are the intersections of pairs of mutually perpendicular tangent planes to a given cyclic cone. 20. Find the condition that five given points may lie on a sphere. 21. What is the surface denoted by the equation where p2 = xa2+yẞ2+zy2, p = xa+yẞ+zy, a, ß, y being given vectors, and x, y, z variable scalars? Express the equation of the surface in terms of p, a, ß, y alone. 22. Find the equation of the cone whose sides bisect the angles between a fixed line and any line, in a given plane, which meets the fixed line. What property of a spherical conic is most directly given by this result? CHAPTER VIII. SURFACES OF THE SECOND ORDER. 248.] THE general scalar equation of the second order in a vector p must evidently contain a term independent of p, terms of the form S.apb involving p to the first degree, and others of the form S.apbpc involving p to the second degree, a, b, c, &c. being constant quaternions. Now the term S.apb may be written as SpV (ba), or as S.(Sa + Va)p(Sb+Vb) = SaSpTB+ SB SpTa+S.pTbTa, each of which may evidently be put in the form Syp, where y is a known vector. Similarly the term S.apbpc may be reduced to a set of terms, each of which has one of the forms Ap2, (Sap)2, Sap SBP, the second being merely a particular case of the third. Thus (the numerical factors 2 being introduced for convenience) we may write the general scalar equation of the second degree as follows: 22.SapSBp+ Ap2+2Syp = C. (1) 249.] Change the origin to D where OD 8, then p becomes p+8, and the equation takes the form 22.Sap SBp+ Ap2 + 2 (Sap Sẞò + Sẞp Sad) + 2 A Sop + 2 Syp ρ from which the first power of p disappears, that is the surface is referred to its centre, if Σ(aSẞò+ẞSad) + Aồ + y = 0, (2) * For S.apbpc=S.capbp =S.a'pbp = (2Sa′Sb— Sa’b) p2 + 2Sa'p Sbp; and in particular cases we may have Va' Vb. |