With a change of sign in the imaginary part, this will represent Thus, as the student will easily find by trial, ẞ and y form with a a rectangular system. But for all that the system of principal vectors of p, viz. a, B±y√1 does not satisfy the conditions of rectangularity. In fact we see by the above values of ẞ and y that S. (B+y√−1)(B−y √√ − 1) = ẞ2 + y2 = − 2 (y2 + z2). It may be well to call the student's attention at this point to the fact that the tensors of these imaginary vectors vanish, for T2 (B ± √ √ − 1) = − S (B±y√−1)(B±y √√ − 1) = y3 − ß2 = 0. This gives a simple example of the new and very curious modifications which our results undergo when we pass to Bivectors; or, more generally, to Biquaternions. As a pendant to the last problem we may investigate the relation of two vector-functions whose successive application produces rotation merely. since each of these functions is evidently self-conjugate. This shews that the pure parts of the strains and x are the same, which is the sole condition. One solution is, obviously, i. e. each of the two is itself a rotation; and a new proof that any number of successive rotations can be compounded into a single one may easily be given from this. But we may also suppose either of , X, suppose the latter, to be self-conjugate, so that or x = x = x 4'4 = x2, which leads to previous results. EXAMPLES TO CHAPTER X. 1. If a, ẞ, y be a rectangular unit system 7-1 S. VapaVߢßVy$y = — mS. ẞp' ̄1aS. ẞ (-) a, and therefore vanishes if be self-conjugate. State in words the theorem expressed by its vanishing. 2. With the same supposition find the values of Also of ΣΤ. Γαφα. Γβφβ and of ΣS. Ταφανβφβ. Σ. ααφα. 3. When are two simple shears commutative ? 1 4. Expanded in powers of 4, and reduce the result to 6. Why cannot we expand p' in terms of 4o, 4, 42? 7. Express Vpop in terms of p, op, op, and from the result find the conditions that op shall be parallel to p. 8. Given the coefficients of the cubic in o, find those of the cubics in 2, 3, &c. p". 12. The cubics in 44 and 4 are the same. in connection with 13. Find the unknown strains and x from the equations $+x=w, 14. Shew that the value of V (paxa + Þßxß +7x7) is the same, whatever rectangular unit system is denoted by a, ß, y. 15. Find a system of simple shears whose successive application results in a pure strain. 16. Shew that, if be self-conjugate, and έ, n two vectors, the two following equations are consequences one of the other :- 17. Shew that in general any self-conjugate linear and vector function may be expressed in terms of two given ones, the expression involving terms of the second order. Shew also that we may write • + z = a (w + x)2 + b (w + x) (w + y) + c (w + y)2, where a, b, c, x, y, z are scalars, and ☎, w the given functions. What character of generality is necessary in and w? How is the solution affected by non-self-conjugation in one or both? 18. Solve the equations: (b) ap + pẞ=Y, (c) p + apßaß, (d) apa1+ẞpB¬1=ypy ̄', (e) apßp=papß. APPENDIX. WE have thought it would be acceptable to many students. if we should give as an Appendix a brief, and in some cases even a detailed, solution of the most important and most difficult of the ADDITIONAL EXAMPLES. In doing so, we would add as a word of advice, that our solutions be employed simply for the purpose of comparison with those which shall occur to the student himself. CHAP. II. Ex. 4. If AB=a, BC= ß, AP = ma, AP' = m'a, BQ =mß, &c.; then gives ma+x {(1 − m) a + mß} = m'a + x' { (1 − m') a + m'ß}, whence x=m', and PE=m'PQ. Ex. 6. ABCD is a quadrilateral; AB = a, AC =ẞ, AD = y, AP = ma, BQ = m (ẞ − a), &c. The condition PQ+ RS = 0 gives (1 − m) a + m (ẞ − a) + (1 − m) (y − B) — my = 0, or - (1 − 2m) (a − B + y) = 0 ; an equation which is satisfied either when 1-2m=0, or when a-B+y=0. The former solution is Ex. 5; the latter gives ABCD a parallelogram. Ex. 10. Let a, b, c be the points in which the bisectors of the exterior angles at A, B, C meet the opposite sides. Let unit T. Q. 14 |