FINITE arithmetical differences have proved remarkably successful in dealing with differential equations ; for instance, approximate particular solutions of the equation for the diffusion of heat crO/dx" = dd/dt can be obtained quite simply and without any need to bring in Fourier analysis. An example is worked out in a paper published in Phil. Trans. A, Vol. 210*. In this book it is shown that similar methods can be extended to the very complicated system of differential equations, which expresses the changes in the weather. The fundamental idea is that atmospheric pressures, velocities, etc. should be expressed as numbers, and should be tabulated at certain latitudes, longitudes and heights, so as to give a general account of the state of the atmosphere at any instant, over an extended region, up to a height of say 20 kilometres. The numbers in this table are supposed to be given, at a certain initial instant, by means of observations.
1. Introduction.— 1·0. The object of this paper is to develop methods where by the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. Though very different in method, it is in purpose a continuation of a former paper by the author, on a “Freehand Graphic Way of Determining Stream Lines and Equipotentials” (‘Phil. Mag.,’February, 1908; also ‘Proc. Physical Soc.,’ London, vol. xxi.). And all that was there said, as to the need for new methods, may be taken to apply here also. In brief, analytical methods are the foundation of the whole subject, and in practice they are the most accurate when they will work, but in the integration of partial equations, with reference to irregular-shaped boundaries, their field of application is very limited.
(Abstract.) In order to deal with irregular boundaries, analysis is replaced by arithmetic, continuous functions are represented by tables of numbers, differentials by central differences. Problems then fall into two classes:-(A) The relation between the equation obtaining throughout the body, and the boundary condition is such that the integral can be stepped out from a boundary. This class includes equations of all orders and degrees.
354sodium hydrate. Acids were also used for dissolving the metallic splash, with negative results.If the xenon is present as a metallic compound, it must therefore go into solution as a compound on dissolution. Silicon hydrides, when treated with sodium hydrate, gives sodium silicate; it may be that xenon behaves in a similar manner, and that a xenate of sodium is formed.
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