Chapter III.
The Hydrodynamics of Analytical Theory.
§ 57. Introductory.—The analytical treatment of hydrodynamic problems commonly involves an extensive application of the higher mathematics, the classic methods being those of Euler, Lagrange, Stokes, and others.
The importance and bearing of the mathematical demonstrations, in connection with the subject of the present work, is comparatively limited, but many of the results are of great consequence; the present exposition has therefore been restricted to a brief indication of the mathematical method, and a digest of those results which, from the present standpoint, are of the greatest interest. Where it has been found possible, a simple physical demonstration is given; in many cases the results of established investigation are taken for granted.
The present discussion opens with a recapitulation of the physical properties of fluids, which may be taken as a concise re-statement of essential definitions, sufficient to render back reference unnecessary. The chapter concludes with a critical argument on the practical deficiencies of the Eulerian and Lagrangian method, and on the theory of Discontinuous Flow.
The hypothesis of the initial discussion is strictly that of an inviscid fluid, and in general the condition of incompressibility is assumed. Up to a certain point the mathematical treatment, as usually applied, takes cognisance of compressibility, but generally speaking, the tangible results, so far as they concern our present subject, relate to the simpler conditions.
§ 58. Properties of a Fluid.—All fluids are characterised by certain definite physical properties. The property that may be
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