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 also represents a cone of the second order; A is a focal line, and 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, B, 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. ρ 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 ρ Sp V (ba), or as S.(Sa+ Ta) (SB+T6) = Sa SpTo+SB SpTa+S.pFbTa, 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+d, and the equation takes the form 22. Sap SBp+ Ap2 + 22(SapSB8+ Sẞp Sad) + 2 A Sop+2 Syp +22. Sad Sẞ8+ Aò2 + 2 Syò− C = 0 ; from which the first power of p disappears, that is the surface is referred to its centre, if * For S.apbpc=S.capbp=S.a'pbp = (2Sa′Sb — Sa’b)p2 + 2 Sa′p Sbp; and in particular cases we may have Va'= Vb. a vector equation of the first degree, which in general gives a single definite value for 8, by the processes of Chapter V. [It would lead us beyond the limits of an elementary treatise to consider the special cases in which (2) represents a line, or a plane, any point of which is a centre of the surface. The processes to be employed in such special cases have been amply illustrated in the Chapter referred to.] With this value of 8, and putting DC-2Syò-Ad2-22. Sad Sẞd, the equation becomes 22.Sap SBp+ Ap2 = D. If D= 0, the surface is conical (a case treated in last Chapter); if not, it is an ellipsoid or hyperboloid. Unless expressly stated not to be, the surface will, when D is not zero, be considered an ellipsoid. By this we avoid for the time some rather delicate considerations. By dividing by D, and thus altering only the tensors of the constants, we see that the equation of central surfaces of the second order, referred to the centre, is (excluding cones) 22(Sap Sẞp)+gp2 = 1. 250.] Differentiating, we obtain or 2 { Sadp SBp+ Sap Sẞdp} +2gSpdp = 0, S.dp{(aSBp+BSap)+gp} = = 0, and therefore, by § 137, the tangent plane is (3) (4) And, as 1 (being perpendicular to the tangent plane, and satisfying its equation) is evidently the vector-perpendicular from the origin on the tangent plane, v is called the vector of proximity. 251.] Hamilton uses for v, which is obviously a linear and vector function of p, the notation pp, & expressing a functional operation, as in Chapter V. But, for the sake of clearness, we will go over part of the ground again, especially for the benefit of students who have mastered only the more elementary parts of that Chapter. We have, then, p = 2(a83p+BẞSap)+gp. With this definition of p, it is easy to see that (a.) $(p+o) = pp+pr, &c., for any two or more vectors. (b.) (xp) = xpp, a particular case of (a), a being a scalar. = = (dp). 2 (d.) Sopp = (Sao SBp+$30 Sap) +gSpσ = Spoσ, or is, in this case, self-conjugate. This last property is of great importance. 252.] Thus the general equation of central surfaces of the second degree (excluding cones) may now be written Differentiating, = Sppp 1. Sappp+Spdopp = 0, (1) which, by applying (c.) and then (d.) to the last term on the left, gives 2 Sppdp = 0, and therefore, as in § 250, though now much more simply, the tangent plane at the extremity of p is This is therefore the equation of the plane of contact of tangent planes drawn from A. 253.] To find the enveloping cone whose vertex is A, notice that where p is any scalar, is the equation of a surface of the second order touching the ellipsoid along its intersection with the plane. If this pass through A we have (Sapa-1)+p(Sapa +1)2 = 0, and p is found. Then our equation becomes (1) (Sppp-1) (Sapa-1)-(Sppa-1)2 = 0, ............. which is the cone required. To assure ourselves of this, transfer the origin to A, by putting p+a for p. The result is, using (a.) and (d.), (Sppp+2Sppa+Sapa—1) (Sapa−1)—(Sppa+Sapa− 1)2 = 0, Sppp (Sapa-1)-(Sppa)2 = 0, or which is homogeneous in Tp2, and is therefore the equation of a cone. |