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and consequently at the surface in question, where ƒ = 0,

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df

dx

These values substituted in (6) give when we replace

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which may always be satisfied by a proper determination, of one of the constants λ and μ, 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

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provided we determine the constant quantities therein contained by means of the equations

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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, since, from the results thus obtained similar ones relative to the cases just enumerated may be found without the least difficulty.

Instead now 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 a. Then it is clear that the equation (2) and condition to which the fluid is subject,

will still remain satisfied. Moreover, if x', y', z' are now referred to three axes fixed in space, we shall have

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and if X' represents the co-ordinate of the centre of the ellipsoid referred to the fixed origin, we shall have

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Adding now to the terme due to the additional velocity, the expression, (3) will then become

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and the velocities of any point of the fluid will be given, by means of the differentials of this last function. But and its differentials evidently vanish at an infinite distance from the solid, where ƒ= ∞; and consequently, the case now under consideration is that of an indefinitely extended fluid, of which the exterior limits are at rest, whilst the parts in the vicinity of the moving body are agitated by its notions.

It will now be requisite to determine the pressure p at any point of the fluid mass. But, by supposing this mass free from all extraneous action, V= 0, and if the excursions of the solid are always exceedingly small, compared with its dimensions, the last term of the second member of the equation (1) may evidently be neglected, and thus we shall have, without sensible error,

-P_dp
---- i.e.
P dt'

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or, by substitution from the last value of p,

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Having thus ascertained all the circumstances of the fluid's motion, let us now calculate its total action upon the moving

solid. Then the pressure upon any point on its surface will be had by making ƒ= 0 in the last expression, and is

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Hence we readily get for the total pressure on the body

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v representing the volume of the body, p" the pressure on that side where x is positive, p, the pressure on the opposite side, and do an element of the principal section of the ellipsoid perpendicular to the axis of x.

If now we substitute for μ its value given from (8), the last expression will become

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Having thus the total pressure exerted upon the moving body by the surrounding medium, it will be easy thence to determine the law of its vibrations when acted upon by an exterior force proportional to the distance of its centre from the point of repose. In fact, let p, be the density of the body, and, consequently, p,v its mass, gX' the exterior force tending to decrease X'. Then by the principles of dynamics,

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If, now, in the formula (10) we substitute for Aλ its value drawn from (9), the last equation will become

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which is evidently the same as would be obtained by supposing the vibrations to take place in vacuo, under the influence of the given exterior force, provided the density of the vibrating body were increased from

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We thus perceive, that besides the retardation caused by the loss of weight which the vibrating body sustains in a fluid, there is a farther retardation due to the action of the fluid itself; and this last is precisely the same as would be produced by augmenting the density of the body in the proportion just assigned, the moving force remaining unaltered.

When the body is spherical, we have a'b'c', and the proportion immediately preceding becomes very simple, for it will then only be requisite to increase p, the density of the body, or half the density of the fluid, in order to have the

by 2/2,
correction in question.

The next case in point of simplicity is where a' c′; for then

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If a>', or the body is an oblate spheroid vibrating in its equatorial plane, the last quantity properly depends on the circular arcs, and has for value

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If, on the contrary, a'<b', or the spheroid is oblong, the value of the same integral is

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Another very simple case is where c', for then the first of the quantities (12) becomes, if a' > b',

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By employing the first of the four expressions immediately preceding, we readily perceive that, when an oblate spheroid vibrates in its equatorial plane, the correction now under consideration will be effected by conceiving the density of the body augmented from

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When b' is very small compared with a', or the spheroid is very flat, we must augment the density

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and we thus see that the correction in question becomes less in proportion as the spheroid is more oblate.

In what precedes, the excursions of the body of the pendulum are supposed very small compared with its dimensions. For if this were not the case, the terms of the second degree in the equation (1) would no longer be negligible, and therefore the foregoing results might thus cease to be correct. Indeed, were we to attend to the term just mentioned, no advantage would even then be obtained; for the actual motion of the fluid where the vibrations are large will differ greatly from what would be assigned by the preceding method, although this method consists in satisfying all the equations of the fluid's

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