Bulletin of the American Mathematical Society Volume 9

Bulletin of the American Mathematical Society Volume 9

List price: US$8.62

Currently unavailable

Add to wishlist

AbeBooks may have this title (opens in new window).

Try AbeBooks

Description

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. 1903 edition. Excerpt: ...surface into its reciprocal with respect to the quadric A. f These theorems are given by Buchheim, Proc. L. M. S., vol. 16, p.15. tThis is a more general statement of a theorem contained in a note in the Messenger of Math., vol. 29, p. 191. If we are reciprocating with respect to the surface x + x + xl + x = Q, we have A = E and the transformation is y=Qx. 8. We shall consider next in some detail the case when JP = 0 and the screw associated with the motion degenerates into a line. We cannot immediately apply the formulas for polar screws because the inverse matrices do not now exist. It is convenient to use two matrices in connection with a line. Let 0--r +q p + )' 0--p' q-q' + p' 0 r--p--q--r 0 where p, q, r, p, q, r' are the coordinates of the line; then PP' s _ (jyp' + qq' +rr') E = 0. Hence if Q, Q' are the matrices associated with another line, the two lines intersect if PQ' + QP' = 0. The lines are polar if Q = AP'A, q = A-lPAl (since A = 1), for then the pole of any plane through one line lies on the other. Hence the line P touches the absolute if it cuts Q, i. e., if P-'P.l-1 + AP'AP' = 0. For any other line we shall take this expression to be equal to--1. This must not be confused with Frobenius's notation for conjugate matrices; in the case of a skew matrix P the conjugate is simply P. To find the distance from a point x to a line P we notice that the plane Px cuts the polar line Q in the point Q'Px and the distance between this and x is xA Q'Px p = sin-1 V-xPAlPx. The distance between x and its displaced position given by the transformation x' = x-f e-4-lPa: is V(eA-lPx)A(eA-lPx) = eV-xPA'Px = esin p; hence e is the small angle of rotation about P. As in the case of two dimensions, so also here the general finite displacement...show more

Product details

  • Paperback | 210 pages
  • 189 x 246 x 11mm | 386g
  • Rarebooksclub.com
  • English
  • Illustrations, black and white
  • 1236977149
  • 9781236977144