to be perfectly smooth, the ratio ought to be independent of the magnitude of the sphere. In the imperfect theory of friction in which the friction of the fluid on the sphere is taken into account, while the equal and opposite friction of the sphere on the fluid is neglected, it is shewn that the arc of oscillation is diminished, while the time of oscillation is sensibly the same as before. But when the tangential action of the sphere on the fluid, and the internal friction of the fluid itself are considered, it is clear that one consequence will be, to speak in a general way, that a portion of the fluid will be dragged along with the sphere. Thus the correction for the inertia of the fluid will be increased, since the same moving force has now to overcome the inertia of the fluid dragged along with the sphere, and not only, as in the former case, the inertia of the sphere itself, and of the fluid pushed away from before it, and drawn in behind it. Moreover the additional correction for inertia must depend, speaking approximately, on the surface of the sphere, whereas the first correction depended on its volume, and thus the effect of friction in altering the time of oscillation will be more conspicuous in the case of small, than in the case of large spheres, other circumstances being the same. The correction for inertia, when friction is taken into account, will not, however, depend solely on the magnitude of the sphere, but also on the time of oscillation. With a given sphere it will be greater for long, than for short oscillations. [From the Transactions of the Cambridge Philosophical Society, Vol. VIII. p. 409.] SUPPLEMENT TO A MEMOIR ON SOME CASES OF FLUID MOTION. Read Nov. 3, 1846. IN a memoir which the Society did me the honour to publish in their Transactions*, I shewed that when a box whose interior is of the form of a rectangular parallelepiped is filled with fluid and made to perform small oscillations the motion of the box will be the same as if the fluid were replaced by a solid having the same mass, centre of gravity, and principal axes as the solidified fluid, but different moments of inertia about those axes. The box is supposed to be closed on all sides, and it is also supposed that the box itself and the fluid within it were both at rest at the beginning of the motion. The investigation was founded upon the ordinary equations of Hydrodynamics, which depend upon the hypothesis of the absence of any tangential force exerted between two adjacent portions of a fluid in motion, an hypothesis which entails as a necessary consequence the equality of pressure in all directions. The particular case of motion under consideration appears to be of some importance, because it affords an accurate means of comparing with experiment the common theory of fluid motion, which depends upon the hypothesis just mentioned. In my former paper, I gave a series by means of which the numerical values of the principal moments of the solid which may be substituted for the fluid might be calculated with facility. The present supplement contains a different series for the same purpose, which is more easy of numerical calculation than the former. The comparison of the * Vol. VIII. Part 1. p. 105. (Ante, p. 17.) two series may also be of some interest in an analytical point of view, since they appear under very different forms. I have taken the present opportunity of mentioning the results of some experiments which I have performed on the oscillations of a box, such as that under consideration. The experiments were not performed with sufficient accuracy to entitle them to be described in detail. The calculation of the motion of fluid in a rectangular box is given in the 13th article of my former paper. I shall not however in the first instance restrict myself to a rectangular parallelepiped, since the simplification which I am about to give applies more generally. Suppose then the problem to be solved to be the following. A vessel whose interior surface is composed of any cylindrical surface and of two planes perpendicular to the generating lines of the cylinder is filled with a homogeneous, incompressible fluid; the vessel and the fluid within it having been at first at rest, the former is then moved in any manner; required to determine the motion of the fluid at any instant, supposing that at that instant the vessel has no motion of rotation about an axis parallel to the generating lines of the cylinder. I shall adopt the notation of my former paper. u, v, w are the resolved parts of the velocity at any point along the rectangular axes of x, y, z. Since the motion begins from rest we shall have udx+vdy+wdz an exact differential do. Let the rectangular axes to which the fluid is referred be fixed relatively to the vessel, and let the axis of x be parallel to the generating lines of the cylindrical surface. The instantaneous motion of the vessel may be decomposed into a motion of translation, and two motions of rotation about the axes of y and z respectively; for by hypothesis there is no motion of rotation about the axis of . According to the principles of my former paper, the instantaneous motion of the fluid will be the same as if it had been produced directly by impact, the impact being such as to give the vessel the velocity which it has at the instant considered. We may also consider separately the motion of translation of the vessel, and each of the motions of rotation; the actual motion of the fluid will be compounded of those which correspond to each of the separate motions of the vessel. For my present purpose it will be sufficient to consider one of the x. motions of rotation, that which takes place round the axis of z for instance. Let w be the angular velocity about the axis of z, w being considered positive when the vessel turns from the axis of x to that of y. It is easy to see that the instantaneous motion of the cylindrical surface is such as not to alter the volume of the interior of the vessel, supposing the plane ends fixed, and that the same is true of the instantaneous motion of the ends. Consequently we may consider separately the motion of the fluid due to the motion of the cylindrical surface, and to that of the ends. Let . be the part of due to the motion of the cylindrical surface, de the part due to the motion of the ends. Then we shall have Consider now the motion corresponding to a value of p, way. It will be observed that way satisfies the equation, {(36) of my former paper,} which is to satisfy. Corresponding to this value of we have Hence the velocity, corresponding to this motion, of a particle of fluid in contact with the cylindrical surface of the vessel, resolved in a direction perpendicular to the surface, is the same as the velocity of the surface itself resolved in the same direction, and therefore the fluid does not penetrate into, nor separate from the cylindrical surface. The velocity of a particle in contact with either of the plane ends, resolved in a direction perpendicular to the surface, is equal and opposite to the velocity of the surface itself resolved in the same direction. Hence we shall get the complete value of by adding the part already found, namely wxy, to twice the part due to the motion of the plane ends. We have therefore, and ¤=wxy + 24e=24c – wxy, by (1) ............... .(2), (3). Hence whenever either pe or pe can be found, the complete solution of the problem will be given by (2). And even when both these functions can be obtained independently, (2) will enable us to dispense with the use of one of them, and (3) will give a relation between them. In this case (3) will express a theorem in pure analysis, a theorem which will sometimes be very curious, since the analytical expressions for d. and p. will generally be totally different in form. The problem admits of solution in the case of a circular cylinder terminated by planes. perpendicular to its axis, and in the case of a rectangular parallelepiped. In the former case, the numerical calculation of the moments of inertia of the solid by which the fluid may be replaced would probably be troublesome, in the latter it is extremely easy. I proceed to consider this case in particular. Let the rectangular axes to which the fluid is referred coincide with three adjacent edges of the parallelepiped, and let a, b, c be the lengths of the edges. The motion which it is proposed to calculate is that which arises from a motion of rotation of the box about an axis parallel to that of z and passing through the centre of the parallelepiped. Consequently in applying (2) we must for a moment conceive the axis of z to pass through the centre of the parallelepiped, and then transfer the origin to the corner, and we must therefore write w(x - La) (y — 1b) for wxy. In the present case the cylindrical surface consists of the four faces which are parallel to the axis of x, and the remaining faces form the plane ends. The motion of the face xy and the opposite face has evidently no effect on the fluid, so that . will be the part of due to the motion of the face xz and the opposite face. The value of this quantity is given near the middle of page 62 in my We have then by the second of the formulæ (2) former paper. 0 ρηπό/α e-nπb/α cos ηπα the sign Σ, denoting the sum corresponding to all odd integral values of n from 1 to co. This value of expresses completely the motion of the fluid due to a motion of rotation of the box about an axis parallel to that of z, and passing through the centre of its interior. Suppose now the motion to be very small, so that the square of the velocity may be neglected. Then, p denoting the part of the pressure due to the motion, we shall have p=-pdp/dt. Also in finding do/dt we may suppose the axes to be fixed in |