Some Invariants and Covariants of Ternary Collineations (Classic Reprint)

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Excerpt from Some Invariants and Covariants of Ternary Collineations 1. The analytical basis of the present paper is the form of Grassmann's Luckenausdruck which Gibbs called a dyadic. This, as the sequel show, is merely a general bilinear function from which the variables are omitted. It may then represent a collineation or correlation and may be manipulated practically like the ordinary symbolical bilinear form. Starting with this as a basis, the object is in the next place to give an interpretation by means of the invariant theory of various double products suggested by Gibbs and incidentally to obtain some of the properties of the invariants and covariants involved. The field of operation is plane projective geometry and the products are formed according to the combinatory multiplication of Grassmann. Finally, in the third part, there is considered a skew symmetric function of any number of collineations which is called an alternant. It is a combinant, linear in the coefficients of each collineation, and presenting in some ways for functions of two sets of variables properties analogous to those of the expressions resulting from the combinatory multiplication of linear manifolds. Part I. Notation. I. The open product or dyadic. 2. In a space of two dimensions a sum of mixed products of similar construction, each containing a single factor x, may by written in the form A1 x B1 + A2 x B2 + A3 x B3 Where the dot is used to show that the order of multiplication is form left to right. A., B, and x are geometric quantities, points or lines of the plane, and all products are formed according to the combinatory multiplication. Thus may be considered as resulting from the operation of x on the expression A1() . B1 + A2() . B2 + A3() . B3, the operation consisting in placing the variable x in the parentheses. This last expression is an example of what Grassman called an open product. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.