Transactions of the Cambridge Philosophical Society Volume 19

Transactions of the Cambridge Philosophical Society Volume 19

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 edition. Excerpt: ...any family C0r) tf0(r-1)... G0(r) AoW-iBoW A0w 50cp)-i... i0a-i50w A0(r) Bo(r)-1 = 1. This is the only condition to be satisfied by the homogeneous matrices and thus all may be chosen arbitrarily except one, say C0r); this is then determinate. If there are no matrices G0) or as we may say no local matricial singularities, the question is rather more complicated. We may then write the condition A0wl B0w A0w = B0(r) Ao(r)-1 Bo(r)-1 Ao(r) B02)... AJ. B0(v). Suppose Aq(r), B0(i) (i=l, 2...p--1) to have been arbitrarily chosen and let JS0 stand for B0(r) Ao(r)'1 Bo(r)-1 A0(r)...A0- so that Ao-1 B0w AQw = EoBo&K The multipliers of JL0(P, _1 B0p) Aoiv) are the same as those of B0(p). Hence B0(p) must be so chosen that its multipliers are the same as those of EQB0ip). The determinant of Bo(p)-6 must then be identically equal to that of JEoBo-0 or of Bo-dEo"1 since the determinant of E0 is clearly unity. The order of the determinant being m we have here m +1 equations to be satisfied two of which are identities, namely those which arise from the term in 0m and the absolute term. The other m--1 are linear in the constituents of any one row or column in Bq&k Thus we may take arbitrarily the constituents in all the rows of B0p) but one, and one even in that row, the others being then determinate. Let the multipliers be 6ly 62... 6mt Then we may suppose B0p) in its canonical form to be x = di%i (i = 1, 2... m), and E0B0M in its canonical form X/ = 0X; (i = 1, 2... ra). Then A0M must be X/ = (i = l, 2...m), where x, ... ATM are arbitrary quantities. This investigation will need to be modified if the multipliers of B0M are not all simple and of the first order, but in any case thshow more

Product details

  • Paperback | 112 pages
  • 189 x 246 x 6mm | 213g
  • English
  • Illustrations, black and white
  • 1236917421
  • 9781236917423