surface of the solid. The most obvious way of doing this is, first to calculate p from the formula

and then to find the resultant force and couple due to the pressure p acting on the various elements dS of the surface by the ordinary rules of Statics. We will work out the result for the simple case of the sphere, starting from the value of given by (12). Since the origin to which is there referred is in motion parallel to OX with velocity V, whereas in (18) the origin is supposed fixed, we. must write, instead of do,

dt'

The whole effect of the fluid pressure evidently reduces to a force in the direction OX. The value of p at the surface of the sphere is

the remaining terms being the same for surface-elements in the positions and π-0, and therefore not affecting the final result. Hence if V be constant, the pressures on the various elements of the anterior half of the sphere are balanced by equal pressures on the corresponding elements of the posterior half; but when the motion of the sphere is being accelerated there is an excess of pressure on the anterior, and a defect of pressure on the posterior half. The reverse holds when the motion is being retarded. The total effect in the direction of V, is

which is readily found to be equal to - πрa3

if M' denote the mass of fluid displaced by the sphere.

If we suppose that the sphere started from rest under the action of a force X constant in direction, so that the centre moves in a straight line, the equation is

The sphere therefore behaves exactly as if its inertia were increased by half that of the fluid displaced, and the surrounding fluid were annihilated.

We have assumed throughout the above calculation that the motion of the sphere is rectilinear. It is not difficult to extend the result to the case where the motion is of any kind whatever. This is effected however more simply by the method of the next article.

The same method can be applied with even greater ease to the case of a long circular cylinder, for which the value of was obtained in Art. 86. It appears that the effect of the fluid pressure is in that case to increase the inertia of the cylinder by that of the fluid displaced exactly.

The practical value of these results, and of similar more general ones to be obtained below, is discussed in note (E).

106. The above direct method of calculating the forces exerted by the fluid on the moving body would, however, in most cases prove exceedingly tedious. This difficulty may be avoided by a method, first used by Thomson and Tait*, which consists in treating the solid and the fluid as forming together one dynamical system, into the equations of motion of which the mutual reactions of the solid and the fluid of course do not enter. As a

# Natural Philosophy, Art. 331.

very simple example of this method we may take the case of the rectilinear motion of a sphere, which has been already investigated otherwise. By the formula of Art. 65, the kinetic energy of the fluid

by (12), or finally M V'. Hence the total energy of the system

must be equal to XV, the rate at which the impressed force X does work. Discarding the common factor V we are led again to the equation (19).

107. In the general case the motion of the fluid at any instant depends, as we saw in Art. 100, only on the values of the quantities u, v, w, p, q, r used to express the motion of the solid; so that the whole dynamical system is virtually one of six degrees of freedom, although it differs in some respects from the kind of system ordinarily contemplated in Dynamics. Thomson* and Kirchhoff† have independently shewn how a system of the peculiar kind here considered may be brought under the application of the ordinary methods of that science. We shall, in what follows, adopt Thomson's procedure, with some modifications.

Whatever be the motion of the fluid and solid at any instant, we may suppose it produced instantaneously from rest by the

*Phil. Mag. November, 1871.

+ Crelle, t. 71. See also Vorlesungen über Math. Physik. Mechanik. c. 19.

action of a properly chosen set of impulsive forces applied to the solid. This set, when reduced after the manner of Poinsot to a force and a couple whose axis is parallel to the line of action of the force, constitute what Thomson calls the 'impulse' of the motion at the instant under consideration. We proceed to shew that when no external impressed forces act the impulse is constant in every respect throughout the motion.

108. The moment of momentum of a spherical portion of the fluid about any line through its centre is zero; for this portion may be conceived as made up of circular rings of infinitely small section having this line as a common axis, and the circulation in each such ring is zero.

In the same way the moment of momentum of a portion of the fluid bounded by two spherical surfaces about the line joining the centres is zero.

The moment of the impulse at any instant about any line is equal to the corresponding moment of momentum at that instant of the whole matter contained within a spherical surface having its centre in that line and enclosing the moving solid; for if we suppose the motion generated instantaneously from rest, the only forces which, besides those constituting the impulse, act on the mass in question are the impulsive pressures on the spherical boundary. Since these act in lines through the centre, they do not affect the moment of momentum.

It is, as was pointed out in Art. 99, immaterial whether we simply suppose the fluid to extend to infinity and to be at rest there, or whether we suppose it contained in an infinitely large fixed rigid vessel which is infinitely distant in all directions from the moving solid. The motion of the fluid within a finite distance of the solid, and therefore the forces exerted by it on the latter, are the same in the two cases. If we now suppose the infinite containing vessel to be spherical in shape, and to have its centre at any point P within a finite distance of the solid, the moment of momentum of the included mass about any line through P is, as we have just seen, equal to the moment of the impulse about the same line. The same reasoning shews that if there be no

external impressed forces this moment of momentum is constant throughout the motion. Hence the moment of the impulse about any line through P is constant. Since in this argument P may be any point within a finite distance of the solid, it follows that the moment of the impulse about any line whatever is constant. This cannot be the case unless the impulse is itself constant in every respect.

We see in the same way that if any external impressed forces act on the solid, the moment of the impulse about any line is increasing at any instant at a rate equal to the moment of these forces about the same line.

The above are somewhat modified proofs of theorems first given by Thomson*. It should be noticed that the reasoning still holds when the single solid is replaced by a group of solids, which may moreover (if of invariable volume) be flexible instead of rigid, and even when these solids are replaced by portions of fluid moving rotationally.

109. The 'impulse' then varies in consequence of the action of the external impressed forces in exactly the same way as the momentum of any ordinary dynamical system does. To express this result analytically let §, n, ; λ, μ, v denote the components of the force- and couple-constituents of the impulse; and let X, Y, Z; L, M, N designate in the same manner the system of external impressed forces. The whole variation of §, n, y, &c., due partly to the motion of the axes to which these quantities are referred, and partly to the action of the forces X, Y, Z., &c., is then given by the formula+: