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RESEARCHES ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA•.
From the Transaction* of tht Royal Soctety <.* Edinburgh-
RESEARCHES ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA.
Probably no department of Analytical Mechanics presents greater difficulties than that which treats of the motion of fluids; and hitherto the success of mathematicians therein has been comparatively limited. In the theory of waves, as presented by MM. Poisson and Cauchy, and in that of sound, their success appears to have been more complete than elsewhere; and if to these investigations wo join the researches of Laplace concerning the tides, we shall have the principal important applications hitherto made of the general equations upon which the determination of this kind of motion depends. The same equations will serve to resolve completely a particular case of the motion of fluids, which is capable of a useful practical application; and as I am not aware that it has yet been noticed, I shall endeavour, in the following paper, to consider it as briefly as possible.
In the Case just alluded to, it is required to determine the circumstances of the motion of an indefinitely extended nonelastic fluid, when agitated by a solid ellipsoidal body, moving parallel to itself, according to any given law, always supposing the body's excursions very small, compared with its dimensions. Prom what will he shown in the sequel the general solution of this problem may very easily be obtained. But as the principal object of our paper is to determine the alteration produced in Ihe motion of a pendulum by the action of the surSGunding medium, we have insisted more particularly on the case where the ellipsoid moves in a right line parallel to one of its axes, and have thence proved, that, in order to obtain the correct time of a pendulum's vibration, il will not be sufficient merely to allow for the loss of weight caused by the fluid medium, but that it will likewise be requisite to conceive the density of the body augmented by a quantity proportional to the density of this fluid. The value of the quantity last named when the body of the pendulum is an oblate spheroid vibrating in its equatorial plane, has been completely determined, and, when the spheroid becomes a sphere, is precisely equal to hall the density of the surrounding fluid. Hence in this last case we shall have the true time of the pendulum's vibration, if we suppose it to move in vaato, and then simply conceive its m«»s augmented by half that of an equal volume of the fluid, whilst the moving force with which it is actuated is diminished by the whole weight of the same volume of fluid.
We will now proceed to consider a particular case of the motion of a non-elastic fluid over a fixed obstacle of ellipsoidal figure, and thence endeavour to find the correction necessary to reduce the observed length of a pendulum vibrating through exceedingly small arcs in any indefinitely extended medium to its true length in vacuo, when the body of the pendulum is a solid ellipsoid. For this purpose we may remark, that the equations, of the motion of a homogeneous non-clastic fluid are
Vide M4c. Cel. lav, in Ch. 8, No. 33, where <J> is such a function of the co-ordinates a;, y, z of any particle of the fluid mass, and of the time t that the velocities of this particle in the directions of and tending to increase the co-ordinates %, y, and z
shall always be represented by -^, ~-, and ~ respectively.
Moreover, p represents the fluid's density, p its pressure, and Va function dependent upon the various forces which act upon he fluid mass.