= Q-2fds S(e2▼u—▼v)▼p—SSƒdse2 (▼p)2. The middle term of this expression may, by the proposition at the beginning of this section, be written 2ƒfƒds & {SV (e2 ▼u)—4πr}, and therefore vanishes. The last term is essentially positive. Thus if u, anywhere differ from u (except, of course, by a constant quantity) it cannot make Q a minimum; and therefore u is a unique solution 2. If two surfaces intersect along a common line of curvature, they meet at a constant angle. 3. By the help of the quaternion formulae of rotation, translate into a new form the solution (given in § 234) of the problem of inscribing in a sphere a closed polygon the directions of whose sides are given. 4. Express, in terms of the masses, and geocentric vectors of the sun and moon, the sun's vector disturbing force on the moon, and expand it to terms of the second order; pointing out the magnitudes and directions of the separate components. 5. If q = r2, shew that 2 dq 11⁄2 = 1 (Hamilton, Lectures, p. 615.) = 2dr = (dr+Kqdrq−1)Sq−1 = § (dr+q−1dr Kq)Sq−1 = 1 = = q ̄1(qdr+Trdrq1) _ dr Uq+Uq-1dr _ dr Kq1+q ̄1dr = = Tq (Uq + Uq ̄1) Tq(1 + Ur). 1+ Ur 2 dq = { dr + V.Vdr'z q} q^' = { dr — V.Vdr's q ̃ 1 } q1 - = drq−1 + V.Vq ̄1V dr (1 + and give geometrical interpretations of these varied expressions for the same quantity. (Ibid. p. 628.) 6. Shew that the equation of motion of a homogeneous solid of revolution about a point in its axis, which is not its centre of gravity, is BV pp-ApVpy, where is a constant. (Trans. R. S. E., 1869.) 7. Integrate the differential equations: where a and b are given quaternions, and 4 and given linear and vector functions. (Tait, Proc. R. S. E., 1870-1.) 8. Derive (4) of § 92 directly from (3) of § 91. 9. Find the successive values of the continued fraction where i and j have their quaternion significations, and ≈ has the values 1, 2, 3, &c. (Hamilton, Lectures, p. 645.) where c is a given quaternion, find the successive values. For what values of c does u become constant? (Ibid. p. 652.) 11. Prove that the moment of hydrostatic pressures on the faces of any polyhedron is zero, (a.) when the fluid pressure is the same throughout, (b.) when it is due to any set of forces which have a potential. 12. What vector is given, in terms of two known vectors, by the relation p1 = (a1+B−1)? · 1 -1 Shew that the origin lies on the circle which passes through the extremities of these three vectors. U 13. Tait, Trans, and Proc. R. S. E., 1870-3. (b.) If S(pv) TNT, (n+3)//frds = -ffr Sp Uvds. (c.) With the additional restriction SSS.Uv (2np+ (n + 3) p2▼).τ ds = 0. (d.) Express the value of the last integral over a nonclosed surface by a line-integral. (e.) -STdp=ƒƒds S.Uvvo, if σ = Udp all round the curve. (f) For any portion of surface whose bounding edge lies wholly on a sphere with the origin as centre Hence shew that the only sets of surfaces which, together, cut space into cubes are planes and their electric images. 15. What problem has its conditions stated in the following six equations, from which §, 7, Ŝ are to be determined as scalar functions of x, y, z, or of p = ix+jy+kz? Shew that they give the farther equations 0 = ▼2§n = ▼2n$ = V2 §§ = ▼2.§n§. Shew that (with a change of origin) the general solution of these equations may be put in the form Sp(p+f)-1p = 1, where is a self-conjugate linear and vector function, and έ, n, Š are to be found respectively from the three values off at any point by relations similar to those in Ex. 24 to Chapter IX. (See Lamé, Journal de Mathematiques, 1843.) 16. Shew that, if p be a planet's radius vector, the potential P of masses external to the solar system introduces into the equation of motion a term of the form S(pv)VP. Shew that this is a self-conjugate linear and vector function of p, and that it involves only five independent constants. Supposing the undisturbed motion to be circular, find the chief effects which this disturbance can produce. 17. In § 405 above, we Va (☎ +n2 ☎) = 0, have the equations Saw 0, a wVia, Ta = 1, Saw = = where 2 is neglected. Shew that with the assumptions wt wit = 2 π a = qßq1, r = ß", w = qrir-1q-1, we have = 0, TB 1, SẞT = 0, = Vß (7+n2T) = 0, provided wSia-w1 = 0. Hence deduce the behaviour of the Foucault pendulum without the x, y, and έ, ŋ transformations in the text. η Apply analogous methods to the problems proposed at the end of § 401 of the text. 18. Hamilton, Bishop Law's Premium Examination, 1862. (a.) If OABP be four points of space, whereof the three first are given, and not collinear; if also OA = a, OB = ß, OP = p ; and if, in the equation the characteristic of operation F be replaced by S, the locus of P is a plane. What plane? (6.) In the same general equation, if F be replaced by V, the locus is an indefinite right line. What line? (c.) If F be changed to K, the locus of P is a point. What point? = (d.) If F be made U, the locus is an indefinite half-line, or ray. What ray? (e.) If F be replaced by T, the locus is a sphere. What sphere? (f) If F be changed to TV, the locus is a cylinder of revolution. What cylinder? (9.) If F be made TVU, the locus is a cone of revolution. What cone? (h.) If SU be substituted for F, the locus is one sheet of such a cone. Of what cone? and which sheet? (2.) If F be changed to VU, the locus is a pair of rays. Which pair? 19. Hamilton, Bishop Law's Premium Examination, 1863. (a.) The equation Spp+a2 = 0 expresses that Ρ and are the vectors of two points P and P', which are conjugate with respect to the sphere p2 + a2 = 0; or of which one is on the polar plane of the other. (b.) Prove by quaternions that if the right line PP', connecting two such points, intersect the sphere, it is cut harmonically thereby. (c.) If p' be a given external point, the cone of tangents drawn from it is represented by the equation, (√pp')2 = a2 (p—p′)2; and the orthogonal cone, concentric with the sphere, by (Spp')2+a2 p2 = 0. (d.) Prove and interpret the equation, T(np-a) Tp-na), if Tp = = Ta. (e.) Transform and interpret the equation of the ellipsoid, T (ip + pk) = k2 — 12. (f) The equation (k2 — c2)2 = (12 + K2) Spp +2 Sɩρкp expresses that p and p' are values of conjugate points, with respect to the same ellipsoid. (9.) The equation of the ellipsoid may also be thus written, Svp = 1, if (x2- 12)2 v = (i−k)2p+2iSkp+ 2kSip. (h.) The last equation gives also, (x2-12)2v = (12 + k2) p2+27ɩpk. |