# Bulletin of the American Mathematical Society Volume 9

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## 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
• English
• Illustrations, black and white
• 1236977149
• 9781236977144