Geometric Properties Completely Characterizing, the Set of All the Curves of Constant, Pressure in a Field of Force (Classic Reprint)
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| Stand: | 2015-08-04 03:50:33 |
Produktbeschreibung
Excerpt from Geometric Properties Completely Characterizing, the Set of All the Curves of Constant, Pressure in a Field of Force In the Princeton Colloquium lectures, 1909, Professor Edward Kasner of Columbia University pointed out several unfinished problems connected with a field of force. He showed that the trajectories whose characteristics he had previously described (Transactions of the American Mathematical Society, Vol. 7, No. 3, pp. 401-424, July, 1906) might be considered as a special case of either of two more general problems: to find curves along which a constrained motion is possible such that the pressure of the moving particle against the curve is (1) proportional to the normal component of the force or (2) constant. The pressure, since the curve is considered smooth, is connected with the normal component of acceleration by the formula. In the case of trajectories a particle moves freely under the action of a force which depends only on the position of the particle; that is, there is no pressure and is obtained when k=0 from P = k N, which represents the first general problem, or when c=0 from P=c, which represents the second general problem. Regarding P=0 as a special case of P = k N, Professor Kasner stated five properties characterizing the system S of curves corresponding to any value of the parameter k. Sarah Elizabeth Cronin in her dissertation, 1917, found geometric properties completely characterizing the system of curves obtained by combining all the systems S. It is my purpose to consider the problem represented by P=c, the problem of curves along which a constrained motion is possible such that the pressure against the curve remains constant. I. I shall prove that the system of curves of constant pressure corresponding to any one value of the parameter c has four properties. Property 1. For any given lineal element the foci of the osculating parabolas of the single infinity of curves determined by the given element lie on a circle passing through the given point. 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.
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