(i.) With the same signification of v, the differential equations of the ellipsoid and its reciprocal become (j.) Eliminate p between the four scalar equations, Sapa, Sẞpb, Syp = c, Sepe. 20. Hamilton, Bishop Law's Premium Examination, 1864. (a.) Let A, B1, A2 B2, ... A,B, be any given system of posited right lines, the 2n points being all given; and let their vector sum, AB = A1B1+A, B2+ ... + A„B„ be a line which does not vanish. Then a point H, and a scalar h, can be determined, which shall satisfy the quaternion equation, AnBn HA1.А1В1+...+HA„.4„B1 = h.AB; namely by assuming any origin O, and writing, h = S AB1+...+AB 1 then this quaternion sum may be transformed as follows, QcQu+CH.AB = (h + CH). AB ; and therefore its tensor is TQc = +CH2).AB, in which AB and CH denote lengths. (c.) The least value of this tensor TQc is obtained by placing the point C at H; if then a quaternion be said to be a minimum when its tensor is such, we may write min. QcQu=h.AB; so that this minimum of Qc is a vector. (d.) The equation TQc = c = any scalar constant > TQH expresses that the locus of the variable point C is a spheric surface, with its centre at the fixed point H, and with a radius r, or CH, such that r.AB = (TQc2-TQu2) = (c2-h2. AB2); so that H, as being thus the common centre of a series of concentric spheres, determined by the given system of right lines, may be said to be the Central Point, or simply the Centre, of that system. (e.) The equation TV Qcc1any scalar constant > TQH represents a right cylinder, of which the radius divided by AB, and of which the axis of revolution is VQc = wherefore this last right line, as being the common axis of a series of such right cylinders, may be called the Central Axis of the system. (f.) The equation SQc = C2 = any scalar constant represents a plane; and all such planes are parallel to the Central Plane, of which the equation is SQc = 0. (9.) Prove that the central axis intersects the central plane perpendicularly, in the central point of the system. (h.) When the n given vectors 4, B1, ... A,B, are parallel, and are therefore proportional to n scalars, b1,... b, the scalar h and the vector QH vanish; and the centre H is then determined by the equation b1.HA1+b2.HA2+...+b„. HA2 = 0, or by the expression, OH= n. b1.042+...+b„.04, ̧ where O is again an arbitrary origin. 21. Hamilton, Bishop Law's Premium Examination, 1860. (a.) The normal at the end of the variable vector p, to the surface of revolution of the sixth dimension, which is represented by the equation (p2 - a2)3 = 27 a2 (p—a)1,... or by the system of the two equations, (a) p2 - a2 = 3 t2 a2, (p—a)2 = t3a2, (a) and the tangent to the meridian at that point, are respectively parallel to the two vectors, so that they intersect the axis a, in points of which the vectors are, respectively, (b.) If dp be in the same meridian plane as p, then t(1-1) (4-1) dp: =3rdt, and spat (c.) Under the same condition, 4-t = (d.) The vector of the centre of curvature of the meridian, at the end of the vector p, is, therefore, (e.) The expressions in Example 38 give v2 = a2 t2 (1−t)2, T2 = a2 t3 (1 − t)2 (4 — t) ; 9 9a2t hence (~—p)2 = 2 a22, and dp2 = ga2t 4 4-t dt2; the radius of curvature of the meridian is, therefore, and the length of an element of arc of that curve is (f.) The same expressions give 4 (Vap)2 = — a1 t3 ( 1 − t)2 (4 −t) ; thus the auxiliary scalar t is confined between the limits = and gives by integration 86 Ta (0-sin 0), if the arcs be measured from the point, say F, for which p = a, and which is common to all the meridians; and the total periphery of any one such curve is = 12πТα. (9.) The value of σ gives 4(o2-a2) = 3a2t(4-t), 16(Vao)2 = — a1 t3 (4 −t)3 ; 4(2-a2)3+27 a2 (Var)2 = 0. ...... (b) (h.) The point F is common to the two surfaces (a) and (b), and is a singular point on each of them, being a triple point on (a), and a double point on (b); there is also at it an infinitely sharp cusp on (b), which tends to coincide with the axis a, but a determined tangent plane to (a), which is perpendicular to that axis, and to that cusp; and the point, say F', of which the vector =- -a, is another and an exactly similar cusp on (b), but does not belong to (a). (i.) Besides the three universally coincident intersections of the surface (a), with any transversal, drawn through its triple point F, in any given direction B, there are always three other real intersections, of which indeed one coincides with F if the transversal be perpendicular to the axis, and for which the following is a general formula: p = Ta.[Ua+ {2SU (ap)+}3 Up]. (j.) The point, say V, of which the vector is p = 2a, is a double point of (a), near which that surface has a cusp, which coincides nearly with its tangent cone at that point; and the semi-angle of this cone is = · π 6 |