HYDRODYNAMICS.) HYDROMECHANICS 455 with similar expressions for the effective inertia parallel to the axes of y and ~. If the outer ellipsoid be indefinitely large, then the effective inertia of the liquid parallel to the axis of x (since A 1 =B 1 = 4 C which, in the case of the sphere, is half the effective inertia of the liquid inside the sphere, since in the sphere A = B = C . For a rotation about the axis of x of the inner ellipsoid, it follows in the same way that the effective inertia of the liquid in the inter space is to the effective inertia of the liquid filling the inner ellip soid in the ratio of the x s f the two motions, which, supposing the outer ellipsoid indefinitely large, o -C ) - ( B - (. )(&* + C- ) + ( A + B + C )(f- - c 2 ) and therefore the effective moment of inertia of the liquid about the axis of j: 4 - (B -C )(6 2 -c 2 ) 2 with similar expressions for the effective moment of inertia about the other axes. Ex. 3. In the case of two spheres and the liquid between, the x s are all zero, and, if the spheres be instantaneously concentric, supposing a the radius of the moving, and a^ that of the fixed sphere. This is a particular case of the confocal ellipsoids, when a = b = c. For then . = and Therefore _1_ _ a? i When the spheres are not concentric, expressions for the effective inertias have been obtained by the method of images by Mr W. M. Hicks (Philosophical Transactions, 1880). The image of a source at P of strength ^ outside a sphere is a source inside the sphere of strength 2fL at a distance > from the centre, a being the radius of the sphere, and a line sink reaching from the image to the centre of line strength -JH ; this combination will be found to produce no flow across the surface of the sphere. Again for a source P of strength /j. inside the sphere, the images will be a source of strength fj^L at the inverse point of P, that i.s, at a distance ^ from the centre, and a line sink - M - thence to infinity. In order that there should be no flow across the spherical boundary another sink of equal strength must exist inside the sphere, and the infinite parts of the line sinks will then cancel. The determination of the f s and x s is a kinematical problem, as yet solved only for the cases we have mentioned, and the discovery ot the solution of fresh problems is at present engaging the attention of mathematicians. Hut supposing them determined for the motion of a body through liquid, then T, the kinetic energy of the body and the liquid, will be a quadratic function of u, v, w, p, q, r ; so that we may put 2T - c llW 2 + c . 2 ,^ + c 33 y.- n ~ + c^p* + c K q* + c ^ + %c n nv+ .... + 2c 5( flr + .... +2c itn + In all twenty-one terms ; and, in order to determine the c s, we mny suppose all the velocities except one or two to vanish, and then we see that where M is the mass of the body, where A is the moment of inertia of the body about the axis of x; these are obtained by supposing all to vanish except u or p. If we suppose all to vanish except v and w, we find and 2T pff<t Similarly the other coefficients may be determined (Kirchhoff, Vorlcsunyen iiber Matkematische 1 hysik, p. 240). In particular cases of symmetry, the coefficients of the products of u, v, w, p, q, r can be made to vanish by a proper choice of axes ; and- in the case of the ellipsoid, the only case for which the coefficients have as yet been determined, 6 2 + c 2 ) + (A + B, + C )(6 2 - c 2 ) while c 12 , . . . vanish, the origin being at the centre of the ellipsoid, and the axes of the ellipsoid its principal axes. In the case o" a sphere of mean density a, projected in infinite liquid of density p, and subject to gravity, the sphere will describe a parabola, with vertical acceleration ~ ? q " Having expressed T nowas a quadratic function of u, v,w,p, q, r, the coefficients being functions of the shape but independent of the position and orientation of the body, the Hamiltonian equations of motion lead to the equations d_i(^T _ dt(du dv _ dtdp r dq dT 7 dw = dT dw For if P de.note the resultant linear impulse in the direction,. fixed in space, whose direction-cosines are I, m, n, then p ,dT dT dT J- "*-; Mtt-r- + n ; du dv dw and, differentiating with respect to the time, since dl dm -. dn 7 = mr-nq. = np-rl. *=lq~mp, dt dt dt f <*Z = I | *(H _ dT dT) dt ( dtdu) T dv + q dw i )tt-/W-JL (Li. ClL + m ^t(dv)-^ + r ^ dT dT for all values of I, m, n. Again, taking a fixed origin, and supposing G the impulsive couple about a straight line through the origin fixed in space whose direction-cosines are I, m, n : G = I rfT _ jZT dw dv ) dT dT -j- ~ x ~r~ du dw/ _ dT dT -, Xj- y-r~ , dr dv * du)
where x, y, ~ are the coordinates of the centre of the body.
