When no external forces act, these equations have the first integrals u§ + vn + w + pλ + qμ + rv = 2T = const., of which the first and second together express the fact that the magnitudes of the force- and couple-constituents of the impulse are constant, and the third the fact that the whole energy of the motion is constant. 110. It remains to express §, n, §, &c. in terms of u, v, w, &c. In the first place let T denote the kinetic energy of the fluid alone, so that where the integration extends over the whole surface of the moving solid. Substituting in this formula the value (2) of 4, we get for 2 an expression of the form 2T Au2 + Bv2 + Cw2 + 2 A'vw + 2B'wu + 2 C'uv = where A, B, C, &c. are certain constant coefficients whose values depend only on the form of the solid, and on the position of the axes of co-ordinates relative to the solid; viz. we have P = p[] x, x ds = -p [[x, (ny — m2) dS, Χι (22), the transformations being effected by the use of (3) and of a particular case of Green's theorem. These expressions for the coefficients are due to Kirchhoff. The kinetic energy, T' say, of the solid alone is also given by a quadratic function of u, v, w, &c., in which however A, B, C are each equal to the mass of the solid, whilst A', B', C', L, M, N, &c. all vanish. The total energy T+T'(=T, say,) of the system is therefore given by a formula of the same form as (21). Except when otherwise indicated we shall suppose A, B, C, &c. to stand for the coefficients in the expression for twice this total energy. 111. The only form of solid for which the coefficients in the expression (21) for 2T have been actually determined is the ellipsoid. We readily find where the notation is the same as in Art. 102, Ex. 3. The values of B, C, Q, R may be written down from symmetry; those of the remaining coefficients are all zero. See Art. 116 (d). Since it appears that if a>b> c, then A <B< C, as might have been anticipated. 112. When in any dynamical system the expression for the kinetic energy in terms of the velocities is known, the values of the component momenta can be derived by a perfectly general process. For this we must refer to books on general Dynamics*. Applied to our case it gives §, n, s, λ, μ, v = dT dT dT dT dT dT ...(23), respectively. These formulæ are readily deduced from those which relate to a perfectly free rigid body by supposing the * See Thomson and Magnetism, Part IV. c. 5. note (C). Tait, Nat. Phil. Art. 313, or Maxwell, Electricity and motion at any instant generated impulsively from rest, and calculating the effect of the impulsive fluid pressures on the surface of the solid. For instance, the resultant impulsive force parallel to x due to this cause is − (Au + C'v + B'w + Lp + L'q + L′′r), (if A, B, C, &c. be supposed for a moment to refer to the fluid only), or da du = momentum of solid parallel to x, (by ordinary Dynamics) and in the same way the rest of the formulæ (23) may be verified. 113. The equations of motion (20) may now be written in We can at once derive some interesting conclusions from these equations, in the case where no external forces act. In the first place Kirchhoff has pointed out that (24) are then satisfied by P, q, r = 0, and u, v, w constant, provided we have i.e. provided the velocity of which u, v, w are the components be in the direction of one of the principal axes of the ellipsoid, Ax2 + By2 + Cz2 + 2A′yz + 2B'zx.+ 20'xy = const. There exist then for every body three mutually perpendicular directions of permanent translation; that is to say, if the body be set in motion parallel to one of these directions, without rotation, and then left to itself, it will continue to move in this manner. It will be seen that these directions are determined by the ratio of the mean density of the solid to the density of the surrounding fluid and by the form of the body's surface. The impulse necessary to produce motion in one of these directions does not in general reduce to a single force; thus if the axes of co-ordinates be chosen, for convenience, parallel to these directions, so that A', B', C'=0, we have corresponding to the motion u alone =Au, n=0, (=0; λ= Lu, μ=0, v=0; so that the impulse consists of a wrench* of pitch L 114. The above, although the simplest, are not the only steady motions of which the body is capable (under the action of no external forces). The instantaneous motion of the body at any instant consists, by a well-known theorem of Kinematics, of a twist about a certain screwt; and the condition that this motion should be permanent is that it should not affect the configuration of the impulse (which is fixed in space) relatively to the body. This requires that the axes of the screw and of the corresponding impulsive wrench should coincide. Since the general equations of a straight line involve four independent constants, this gives four relations to be satisfied by the five ratios u v w : p: q: r. * A 'wrench' is a system of forces supposed reduced after the manner of Poinsot to a force and a couple having its axis in the direction of the force. Its 'pitch' is the line which is the result of dividing the couple by the force. See Ball, Theory of Screws. + A 'twist' is the most general motion of a rigid body, equivalent to a translation parallel to some axis combined with a rotation about that axis. Its 'pitch' is the linear magnitude which is the ratio of the translation to the rotation. Ball, Theory of Screws. There exists then for every body, under the circumstances here considered, a simply-infinite system of possible steady motions. Of these the next in importance to the three motions of permanent translation are those in which the impulse reduces to a couple only. The equations (20) or (24) are satisfied by §, n, 5=0, and λ, μ, v constant, provided If the axes of co-ordinates have the special directions adopted in Art. 113, the conditions &, n, =0 give, us at once u, v, w in terms of p, q, r, viz. И Lp + L'g+ L'r ̧ &c., &c. ......... (26). A Substituting these values in the expressions for λ, μ, v obtained from (23), we find 20 = Pp2 +Qq2 + Rr2 + 2P′qr +2Q'rp + 2R′pq.....(28); the coefficients in this expression being determined by the formulæ These formulæ hold for any case in which the force-constituent of the impulse is zero. Introducing the conditions (25) for steady motion, we have to determine p q r the three equations The form of (29) shews that the line whose direction-ratios are p: qr is parallel to one of the principal axes of the ellipsoid |