configuration such that VT, is negative, the above expression, and therefore à fortiori the part V - To, will assume continually increasing negative values, which can only take place by the system deviating more and more from its equilibrium-configuration. This important distinction between 'ordinary' or kinetic, and 'secular' or practical stability was first pointed out by Thomson and Tait*. It is to be observed that the above investigation presupposes a constant angular velocity (w) maintained, if necessary, by a proper application of force to the rotating solid. When the solid is free, the condition of secular stability takes a somewhat different form, to be referred to later (Chapter XII.). To examine the character of a free oscillation, in the case of stability, we remark that if λ be any root of (10), the equations (8) give where A1, A2, Ar3, ... Arn are the minors of any row in the determinant A, and C is arbitrary. It is to be noticed that these minors will as a rule involve odd as well as even powers of A, and so assume unequal values for the two oppositely signed roots (A) of any pair. If we put = io, the general symbolical value of q, corresponding to any such pair of roots may be written we get a solution of our equations in real form, involving two arbitrary constants K, e; thus† In = Fn (o2). K cos (ot + €) — ofn (o2). K sin (ot + €).. These formulae express what may be called a 'natural mode' of oscillation of the system. The number of such possible modes is of course equal to the number of pairs of roots of (10), i.e. to the number of degrees of freedom of the system. * Natural Philosophy (2nd ed.), Part I. p. 391. See also Poincaré, "Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation," Acta Mathematica, t. vii. (1885), and op. cit. ante p. 141. We might have obtained the same result by assuming, in (5), If έ, n, denote the component displacements of any particle from its equilibrium position, we have Substituting from (13), we obtain a result of the form = P. K cos (ot + €) + P' . K sin (ot † e), = η Q. K cos (ot + €) + Q' . K sin (σt + €), .(15) where P, P', Q, Q', R, R' are determinate functions of the mean position of the particle, involving also the value of o, and therefore different for the different normal modes, but independent of the arbitrary constants K, e. These formulae represent an elliptic-harmonic motion of period 2′′/σ, the directions being those of two conjugate semi-diameters of the elliptic orbit, of lengths (P2 + Q2 + R2)1. K, and (P'2 + Q'2 + R'2). K, respectively. The positions and forms and relative dimensions of the elliptic orbits, as well as the relative phases of the particles in them, are accordingly in each natural mode determinate, the absolute dimensions and epochs being alone arbitrary *. 206. The symbolical expressions for the forced oscillations due to a periodic disturbing force can easily be written down. If we assume that Q1, Q2,... Qn all vary as eat, where σ is prescribed, the equations (5) give, if we omit the time-factors, *The theory of the free modes has been further developed by Rayleigh, "On the Free Vibrations of Systems affected with Small Rotatory Terms," Phil. Mag. (6), t. v. p. 293 (1903) [Papers, t. v. p. 89], for the case where the rotatory coefficients ẞ, are relatively small. The most important point of contrast with the theory of the 'normal modes' in the case of no rotation is that the displacement of any one type is no longer affected solely by the disturbing force of that type. As a consequence, the motions of the individual particles are, as is easily seen from (14), now in general elliptic-harmonic. Again, there are in general differences of phase, variable with the frequency, between the displacements and the force. As in Art. 168, the displacement becomes very great when ▲ (io) is very small, i.e. whenever the 'speed' o of the disturbing force approximates to that of one of the natural modes of free oscillation. When the period of the disturbing forces is infinitely long, the displacements tend to the 'equilibrium-values' as is found by putting σ = σ = 0 in (17), or more simply from the fundamental equations (5). This conclusion must be modified, however, when one or more of the coefficients of stability C1, C2, ... cn is zero. If, for example, c1 = 0, the first row and column of the determinant ▲ (A) are both divisible by A, so that the determinantal equation (10) has a pair of zero roots. In other words we have a possible free motion of infinitely long period. The coefficients of Q2, Q3, ... Qn on the right-hand side of (17) then become indeterminate for σ = 0, and the evaluated results do not as a rule coincide with (18). This point is of importance, because in some hydrodynamical applications, as we shall see, steady circulatory motions of the fluid, with a constant deformation of the free surface, are possible when no extraneous forces act; and as a consequence forced tidal oscillations of long period do not necessarily approximate to the values given by the equilibrium theory of the tides. Cf. Arts. 214, 217. In order to elucidate the foregoing statements we may consider more in detail the case of two degrees of freedom. The equations of motion are then of the forms a1Ÿ1 +191 +ẞq2 = Q1, α 2Ÿ2 +C292 −ẞģ1 = Q2• The equation determining the periods of the free oscillations is а1a ̧λ1 + (α1¤ ̧ + α 2 C1 + (2) λ2 + c1 C2 = 0. For 'ordinary' stability it is sufficient that the roots of this quadratic in λ2 should be real and negative. Since a, a, are essentially positive, it is easily seen that this condition is in any case fulfilled if c1, c2 are both positive, and that it will also be satisfied even when C1, C2 are both negative, provided 32 be sufficiently great. It will be shewn later, however, that in the latter case the equilibrium is rendered unstable by the introduction of dissipative forces. See Art. 316. To find the forced oscillations when Q1, Q2 vary as eat, we have, omitting the Let us now suppose that c2 =0, or, in other words, that the displacement q2 does not affect the value of V-To. We will also suppose that Q2 =0, i.e. that the extraneous forces do no work during a displacement of the type q2. The above formulae then give In the case of a disturbance of long period we have σ =0, approximately, and therefore The displacement q, is therefore less than its equilibrium-value, in the ratio 1:1 +ẞ2/a2C1; and it is accompanied by a motion of the type q, although there is no extraneous force of the latter type (cf. Art. 217). We pass, of course, to the case of absolute equilibrium, considered in Art. 168, by putting ẞ=0*. It should be added that the determination of the 'principal co-ordinates' of Art. 204 depends on the original forms of T and V - To, and is therefore affected by the value of w2, which enters as a factor of To. The system of equations there given is accordingly not altogether suitable for a discussion of the question how the character and the frequencies of the respective principal modes of free vibration vary with w. One remarkable point which is thus overlooked is that types of circulatory motion, which are of infinitely long period in the case of no rotation, may be converted by the slightest degree of rotation into oscillatory modes of periods comparable with that of the rotation. Cf. Arts. 212, 223. To illustrate the matter in its simplest form, we may take the case of two degrees of freedom. If c, vanishes for w=0, and so contains w2 as a factor in the general case, the two roots of equation (20) are approximately, when w2 is small. The latter root makes λ o w, ultimately. 207. Proceeding to the hydrodynamical examples, we begin with the case of a plane horizontal sheet of water having in the undisturbed state a motion of uniform rotation about a vertical axist. The results will apply without serious qualification to the case of a polar or other basin, of not too great dimensions, on a rotating globe. Let the axis of rotation be taken as axis of z. The axes of x and y being now supposed to rotate in their own plane with the prescribed angular velocity w, let us denote by u, v, w the velocities at time t, relative to these axes, of the particle which then occupies the position (x, y, z). The actual velocities of the same particle, parallel to the instantaneous positions of the axes, will wy, v + wx, w, and the accelerations in the same directions will be be u * The preceding theory appeared in the 2nd ed. (1895) of this work. † Sir W. Thomson, "On Gravitational Oscillations of Rotating Water," Proc. R. S. Edin. t. x. p. 92 (1879) [Papers, t. iv. p. 141]. In the present application, the relative motion is assumed to be infinitely small, so that we may replace D/Dt by d/ǝt. Now let zo be the ordinate of the free surface when there is relative equilibrium under gravity alone, so that is as in Art. 26. For simplicity we will suppose that the slope of this surface everywhere very small; in other words, if r be the greatest distance of any part of the sheet from the axis of rotation, w2r/g is assumed to be small. If Zo the usual assumption that the vertical acceleration of the water is small compared with g, the pressure at any point (x, y, z) will be given by + denote the ordinate of the free surface when disturbed, then on whence P - Po = gp (zo + ( − z), 1 др ρ θα w2x -- 1 др рду The equations of horizontal motion are therefore where denotes the potential of the disturbing forces. .(2) .(3) The equation of continuity has the same form as in Art. 193, viz. where h denotes the depth, from the free surface to the bottom, in the undisturbed condition. This depth will not, of course, be uniform unless the bottom follows the curvature of the free surface as given by (1). If we eliminate - from the equations (5), by cross-differentiation, we find or, writing u =əğlət, v=ôŋ/ôt, and integrating with respect to t, .(7) .(8) This is merely the expression of Helmholtz' theorem that the product of the vorticity |