Proceedings of the London Mathematical Society Volume 10 by London Mathematical Society
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Author: London Mathematical Society
Page Count: 58 pages
Published Date: 01 Mar 2012
Publisher: Rarebooksclub.com
Publication Country: Miami Fl, United States
Language: English
ISBN: 9781130230468
File size: 48 Mb
Download Link: Proceedings of the London Mathematical Society Volume 10
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1879 Excerpt: ... here meant an expression--lineo-linear in two sets of such numbers regarded as variables, say X X; fiv /u2, such as an Xjj + aXjfjj-l-cr.212/n1 + a22X, !u, (2). This may be also denoted, for shortness, by the symbol a 12, or even by 12, where the 1, 2 refer to the two sets of variables X, fi. In this case there is one invariant only which allows of the variables being separately transformed, namely, the discriminant, which is got by squaring the form. We have, in fact, by the properties of alternate numbers, (a 12)s =--2 (aua23--a12a21) =--2D, suppose (3). But if the variables are transformed by the same substitution, there is a universal covariant, Xi, --Xs/Uj, which may be denoted by (12), or by (21); for Xjjj--X2fij = /X, --jUjXj, by the property of alternate numbers. If we replace the /i's by x, an ordinary, not an alternate, number, we get a linear function of the X's, whose square vanishes in virtue of the relations (1). Thus we have (a lie)2 = 0, and consequently such a product as (a las) (a ly) must be divisible by (xy) = -ij/2--x$x This is the substance of some fragments found amongst the papers of the late Professor Clifford. The only published explanation of the method with which I am acquainted, is contained in the following sliort paragraph from a letter to Professor Sylvester (see "American Journal of Pure and Applied Mathematics," Vol. i., p. 127): "I consider forms which are linear in a certain number of sets of k variables each. To fix the ideas, suppose k = 2, and that I have altogether 6 sets of 2 variables each, namely, ViV-K -i2s, -;, -, . Suppose the forms are xyzu), (yzvw), (xv), mmi); viz., (x/zu) means an expression separately linear, and homogeneous in the x, the y, the z, the u, and so for th...
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