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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 waves, as presented by MM. Poisson and Cauchy, and in that of sound, then 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 ca9e 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.
Irt 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 produced in the motion of a pendulum by the 8ction of the sursounding 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 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 half 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 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 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-elastic fluid are
Vide M4c. Cel. Liv. lit. Ch. 8. No. 33, where <f> is such a function of the co-ordinates x, 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 af, y, and z
shall always be represented by , , and ^~ respectively.
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; X and p being two arbitrary quantities constant with regard to the variables sc, y, z: and a. b, c being functions of these hama variables, determined by the equations
«W»+/, b*=b'*+f, cW-f/, and 1 ....(4),
in which a, b', 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),
afy ff _ 2m if tut /d'f d°f dy\
d& + df . d?' aFbe dx o?bo \dj? + dtf + dVJ
'a3bc \9j W 2ci \\4jb) \dy) \dz> ]'
* In ray memoir on the Determination of the exterior and interior Attractions of Ellipsoids of Variable Densities', recently communicated to the Cambridge Philosophical Society by Sir Edward Ffkkmck Bbohhead, Baronet, I have given a method by which the general integral of the partial differential equation.
°" fte,> + oV + + dx} + tfV a ~du
mav be expanded in a series of peculiar form, and have thus rendered tho determination of these attractions a matter of comparative facility. The same method applied to the equation (») of the present paper has the advantage of giving an expansion of its general integral, every term of which, besides satisfying this equation, may 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 thought advisable to introdoce into the test a demonstration of this particular case.
1 [Vid. supra, p 185.]
Moreover, by the same means, the last of the equation (4) gives
. d*f tff , cZ8/ ?+
dx* dy* <UJ x' y' z1
~« + CT + 7«
a o c
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 A. and fi that the fluid particles may move along the surface of the ellipsoidal body of which the equation is
But by differentiation, there results
U~ a" + b"- + o"'
and as the particles must move along the surface, it is clear that the last equation ought to subsist, when we change the elements
dx, dy, and az into their corresponding velocities d-r-, ,
and Hence, at this surface
° ~ a!»dx + b*dy+ c* d* """W' But the expression (3) gives generally
-^ = X+u( & + -EL dl it-WL11/ ty-JPtf h\ dx T *\ybc r a*bc dx * dy a'6c dy' dz~ a%c <fe '*