Geometric Properties Completely Characterizing The Set Of All The Curves Of Constant Pressure In A Field Of Force (1922) - Tapa blanda
Reseña del editor
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Reseña del editor
In the Princeton CoU oquiimi lectures, 1909, Professor Edward Kasner of Coltimbia University pointed out several unfinished problems connected with a field of force. He showed that the trajectories whose characteristics he had previously described (T ransactions 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 p= iV. In the case of trajectories a particle moves freely imder the action of a force which depends only on the position of the particle; that is, there is no pressure andF =o. F= ois obtained when k=o from P= kN, which represents the first general problem, or when c= ofrom P=c, which represents the second general problem. Regarding P=={) as a special case oiP =kN Professor Kasner stated five properties characterizing the sjtem Sof oo curves corresponding to any value of the parameter fe. Sarah Elizabeth Cronin in her dissertation, 1917, found geometric properties completely characterizing the system of oo 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 systemS cof oo curves of constant pressure corresponding to any one value of the parameter chas four properties. Property 1. For any given lineal element {x, y, y). the foci of the osculating parabolas of the single infinity of curves determined by the given element He on a
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