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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 wayes, 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 we 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 a8 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. From what will be 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 pro. duced in the motion of a pendulum by the action of the sur. sounding 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, it 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 flaid. The value of the quantity last named when the body of the pendolum is an oblate spberoid vibrating in its equatorial plane, has been coinpletely deterniined, and, when the spheroid becomes a sphere, is precisely equal to half the density of the surrounding fluid. llence in this last case we shall have the true time of the pendulum's vibration, if we suppose it to move in vacuo, and then simply conceive its mass 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 mediuin to its true length in vacun, when the body of the pendulum is a solid ellipsoid. For this purpose we may remark, that the equations of the motion of a homogencous non-clastic fluid are
p_dø. 1 100.06)dbil?
do 10 db a= dat dy + dat
................ (2). Vide Méc. Cél. Liv, ul. Ch. 8, No. 33, where is such a function of the co-ordinates x, y, z of any particle of the fluid mass, and of the time i that the velocities of this particle in the directions of and tending to increase the co-ordinates x, y, and . shall always be represented by any team, and respectively,
x' dy Moreover, p represents the fluid's density, p its pressure, and V a function dependent upon the various forces which act upon he fluid mass.
When the fluid is supposed to move over a fixed solid ellipsoid, the principal difficulty will be so to satisfy the equation (2), that the particles at the surface of this solid may move along this surface, which may always be effected by making
supposing that the origin of the co-ordinates is at the centre of the ellipsoid; 2 and being two arbitrary quantities constant with regard to the variables x, y, z: and a, b, c being functions of these same variables, determined by the equations
in which a, 6', c' are the axes of the given ellipsoid.
To prove that the expression (3) satisfies the equation (2), it may be remarked, that we readily get, by differentiating (3),
d'o. d'$, d¢ _ 24 df _ Mit 070991)
* In my memoir on the Determination of the exterior and interior Attrac. tions of Ellipsoids of Variable Densities', recently communicated to the Cambridge Philosophical Society by Sir EDWARD FFRENCH BROUHEAD, Baronet, I have given a method by which the general integral of the partial differential equation,
at2 y d y el? Vď ľn-sdV of dr, at drit + dret durt u du
may be expanded in a series of peculiar forrn, and have thus rendered the determination of these attractions a matter of comparative facility. The same method applied to the equation (2) of the present paper has the advantage of giving an expansion of its general integral, every term of which, besides satisfying this equation, roay likewise be made to satisfy the condition (6). The formula (3) is only an individual term of the expansion in question. But in order to render the present communication independent of every other, it was thougbt advisable to introduce into the test a demonstration of this particular case.
1 (Vid. supra, p 135]
Moreover, by the same means, the last of the equation (4) gives
which values being substituted in the second member of the preceding equation, evidently cause it to vanish, and we thus perceive that the value (3) satisfies the partial differential equation (2).
We will now endeavour so to determine the constant quantities 1 and 4. that the Auid particles may move along the surface of the ellipsoidal body of which the equation is
But by differentiation, there res
and as the particles must move along the surface, it is clear that the last equation ought to subsist, when we change the elements
do do da, dy, and az into their corresponding velocities and Hence, at this surface
On my do, y do 2 de
die + Çantada ..............(6). But the expression (3) gives generally dos dfue df domy do uz df de 17mabc abc doe dy V abc dz**
and consequently at the surface in question, where / = 0,
de =+ og andre
These values substituted in (6) give when we replac
ole and is with their values at the ellipsoidal surface,
0=+vloemen en te .......(B).
which may always be satisfied by a proper determination of one of the constants , and y, the other remaining entirely arbitrary.
From what precedes, it is clear that the equation (2) and condition to which the fluid is subject may equally well be satisfied by making
provided we determine the constant quantities therein contained by means of the equations
respectively. The same may likewise be said of the sum of the three values of $ before given. However, in what follows, we shall consider the value (3) only, sioce, from the results thus obtained similar ones relative to the cases just enumerated may be found without the least difficulty.
Instead dow of supposing the solid at rest, let every part of the whole system be animated with an additional common velocity - in the direction of the co-ordinate x. Then it is clear that the equation (2) and condition to which the fluid is subject.