EXAMPLES TO CHAPTER V. 1. Solve the following equations: (a.) V.apẞV.ayẞ. (6.) αρβρ = ραρβ. (c.) appẞy. (d.) S.aßp+ ẞSap—aVßp = y. (e.) papßaß. (7.) αρβρ = ρβρα. Do any of these impose any restriction on the generality of a and ẞ? 3. If λ, u, v be any three non-coplanar vectors, and 4 = Γμν.φλ + Γυλ.φμ + Γλμ.φν, shew that q is necessarily divisible by S.λuv. Also shew that the quotient is where Vep is the non-commutative part of pp. where r is a given quaternion, shew that m=Σ(S.a1α2аzS.ẞ3ẞ2ß1) + ΣS(rVα1α2. Vẞ2ẞ1) + SrΣS.aßr-Σ(SarSẞr) + SrTr2, and mp1o=2(Va1a2S.ẞ2310) +ΣV.aV (Vßo.r) + VorSr-VrSor. Lectures, p. 561. shew that the following relations exist among any five quaternions 0 = p(qrst)+q(rstp)+r(stpq)+s(tpqr)+t(pqrs), q(prst)=[rst]Spq−[stp]Srq+[tpr]Ssq−[ prs]Stq. and Elements, p. 492. 7. Shew that if o, be any linear and vector functions, and a, B, y rectangular unit-vectors, the vector 0 =V (φαψα + φβψβ + φγψγ) is an invariant. [This will be immediately seen if we write it in the form 0 =7.φνψρ, which is independent of the directions of a, ß, y. practice to dispense with V.] But it is good The scalar of the same quaternion is also an invariant, and may be written as 8. Shew that if = ppaSap+BSBp+ySyp, where a, ß, y are any three vectors, then 9. Shew that any self-conjugate linear and vector function may in general be expressed in terms of two given ones, the expression involving terms of the second order. Shew also that we may write $ + z = a (∞ + x)2 + b (w + x) (w + y)+c(w+y)2, where a, b, c, x, y, z are scalars, and and w the two given functions. What character of generality is necessary in and o? How is the solution affected by non-self-conjugation in one or both? 10. Solve the equations:· : (a.) q2 = 5qi+10j. (b.) q2 = 2q+i. (c.) qaq = bq+c. (d.) aq=gr=rb. 12. If be self-conjugate, and a, ß, y a rectangular system, 8.Гафафуфу = 0. 13. oy and yo give the same values of the invariants m, m, m2. 14. If ' be conjugate to 4, pp′ is self-conjugate. 15. Shew that (Va0)2 + (Vẞ0)2 + (Vy0)2 if a, ẞ, y be rectangular unit-vectors. 16. Prove that = 202 17. Solve the equations: (a.) 42=; (b.) 4+x=w, | where one, or two, unknown linear and vector functions are given in terms of known ones. (Tait, Proc. R. S. E. 1870-71.) 18. If be a self-conjugate linear and vector function, έ and n two vectors, the two following equations are consequences one of the other, viz. : ૐ Γ.ηφη From either of them we obtain the equation Sp&pn = S*.§p&p2§S3.n&np2n. This, taken along with one of the others, gives a singular theorem when translated into ordinary algebra. What property does it give of the surface S.pppp2p = 1? CHAPTER VI. GEOMETRY OF THE STRAIGHT LINE AND PLANE. 188.] HAVING, in the five preceding Chapters, given a brief exposition of the theory and properties of quaternions, we intend to devote the rest of the work to examples of their practical application, commencing, of course, with the simplest curve and surface, the straight line and the plane. In this and the remaining Chapters of the work a few of the earlier examples will be wrought out in their fullest detail, with a reference to the first five whenever a transformation occurs; but, as each Chapter proceeds, superfluous steps will be gradually omitted, until in the later examples the full value of the quaternion processes is exhibited. 189.] Before proceeding to the proper business of the Chapter we make a digression in order to give a few instances of applications to ordinary plane geometry. These the student may multiply indefinitely with great ease. (a.) Euclid, I. 5. Let a and B be the vector sides of an isosceles triangle; B-a is the base, and Ta= TB. The proposition will evidently be proved if we shew that a(a—ß)−1=Kß(ß—a)-1 (§ 52). a(a-3)-1-(3-a)-1B, This gives (b.) Euclid, I. 32. Let ABC be the triangle, and let where y is a unit-vector perpendicular to the plane of the triangle. l If = 1, the angle CAB is a right angle (§ 74). Hence This is, properly speaking, Legendre's proof; and might have been given in a far shorter form than that above. In fact we have for any three vectors whatever, which contains Euclid's proposition as a mere particular case. (c.) Euclid, I. 35. Let ẞ be the common vector-base of the parallelograms, a the conterminous vector-side of any one of them. For any other the vector-side is a +xß (§ 28), and the proposition appears as ᎢᏤ TVB (a+x3) = TV ßa (§§ 96, 98), which is obviously true. (d.) In the base of a triangle find the point from which lines, drawn parallel to the sides and limited by them, are equal. If a, ẞ be the sides, any point in the base has the vector which bisects the vertical angle of the triangle. This is not the only solution, for we should have written instead of the less general form above which tacitly assumes that 1-∞ and x are positive. We leave this to the student. |