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General equations of the equilibrium of fluids
Sect. I.-Incompressible fluids
III.-Equilibrium of revolving fluids
Motion of Solid Bodies acted on by Fluids.
Sect. I.—Finite vertical oscillations in incompressible fluids
II.—Motion of bodies subject to the action of aeriform fluids
III.-Small oscillations of floating bodies
NORMAL PRESSURE OF FLUIDS.
SECTION I. Normal Pressure of Homogeneous Incompressible Fluids on the Surfaces
of Immersed Solids. LET K denote any indefinitely small element of the surface of a solid immersed in a fluid, and let x represent the depth of K below the surface of the fluid ; then, g denoting the accelerating force of gravity and p the density of the fluid, gpKx will represent the pressure of the fluid on K, and the magnitude of the pressure on the whole surface of the solid will be equal to
gρΣ (Κ.α), where denotes the summation of a series of terms, of which Ku is the general type, the limits of the summation being defined by the boundary of the immersed surface.
Cor. 1. If z denote the depth of the centre of gravity of the surface of the solid below the surface of the fluid, then, A representing the surface of the solid, we know that
(Kx) = AT, and therefore the pressure on A is equal to
gpaz, that is, to the weight of a column of the fluid the base of which is equal to the surface of the solid immersed and the height to the depth of the centre of gravity of the surface of the solid
below the surface of the fluid. This proposition was first enunciated generally by Cotes, in his Hydrostatical and Pneumatical Lectures, p. 37, 3rd edit.*
When the magnitude of the area (A) and the position of its centre of gravity are known, this corollary enables us at once to determine the whole pressure upon it.
COR. 2. Suppose that the area A is divisible into a finite number of areas of known magnitudes, A,, A,, A,,, &c., the positions of the centres of gravity of which are known, then the pressure on (A) will be equal to
gp (A, , + A,, ,, + A, T,, +. ...), where 7, ā,,, . ... denote the depths of the centres of gravity of A,, A,,, A1,.. ...respectively below the surface of the fluid.
Whenever, as is most commonly the case, the position of the centre of gravity of the surface of the immersed solid is not known, and it cannot be subdivided into any finite number of areas of which the centres of gravity are known, the evaluation of the expression gps (K.c) must be effected by means of the Integral Calculus.
We have supposed above that the surface of the fluid is free from pressure: should this not be the case, we must consider x to denote the sum of the depth of K below the surface of the fluid and the length (H) of a vertical column of the fluid which would exert a pressure on its base equal to that exerted upon an equal area in the surface of the fluid. This amounts to supposing the depth at any point of the fluid to be x + H instead of x.
The principles of the determination of the pressure of fluids, on plane surfaces, were first laid down by Stevin. His hydrostatical investigations may be seen in the third volume of his Hypomnemata Mathematica, translated into Latin from the Dutch by Snell and published at Leyden in the year 1608, or in the fourth volume of Les Euvres Mathématiques de Simon Stevin de Bruges, par Albert Girard Samielois, Mathematicien,
* The first edition of these Lectures was published in 1737, by Dr. Smith, Master of Trinity College, Cambridge.