Lanterna Rally

Author: Eric Harold Neville

Publisher: Forgotten Books

Published: 2015-06-25

Total Pages: 84

ISBN-13: 9781330186398

Excerpt from Multilinear Functions of Direction: And Their Uses in Differential Geometry The distinctive feature of this work is that the functions discussed are primarily not functions of a single variable direction but functions of several independent directions. Functions of a single direction emerge when the directions originally independent become related, and a large number of elementary theorems of differential geometry express in different terms a few properties of a few simple functions; since one of the objects of the essay is to emphasise the coordinating power of the theory, the presence of many results with which every reader will be thoroughly familiar calls for no apology. In the applications to the geometry of a single surface two functions thought to be new are described. The first, studied in Section 4, depends on two tangential directions, reduces to normal curvature when these directions coincide, and is called here bilinear curvature. I became acquainted with this function in 1911 and used it in lectures early in 1914. The second, the subject of. Section 6, depends on three directions, and reduces to the cubic function associated with the name of Laguerre; the function is symmetrical, and because the equations of Codazzi can be read as asserting its symmetry I have called the general function the Codazzi function. The theory of multilinear functions doe9 not merely coordinate. It affords simple proofs of the relations between the cubic functions of Laguerre and Darboux (6'231, 6 234) and of formulae (7 242, 7-351, 7-352) for the twist of a family of surfaces, and it leads naturally to expressions (7 241) for the rates of change of the two principal curvatures of a variable member of a family of surfaces at the current point of an orthogonal trajectory of the family, expressions that are interesting because their existence was deduced by Forsyth in 1903 from an enumeration of invariants. 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.