figures: not stable; (b) If the condition of being a figure of revolution is im- Annular posed, without the condition of being an ellipsoid, there is, for probably large enough moment of momentum, an annular figure of equilibrium which is stable, and an ellipsoidal figure which is unstable. It is probable, that for moment of momentum greater than one definite limit and less than another, there is just one annular figure of equilibrium, consisting of a single ring. (c) For sufficiently large moment of momentum it is certain that the liquid may be in equilibrium in the shape of two, three, four or more separate rings, with its mass distributed among them in arbitrary portions, all rotating with one angular velocity, like parts of a rigid body. It does not seem probable that the kinetic equilibrium in any such case can be stable. under con remain (d) The condition of being a figure of equilibrium being still unless imposed, the single-ring figure, when annular equilibrium is straint to possible at all, is probably stable. It is certainly stable for very symmetri large values of the moment of momentum. (e) On the other hand let the condition of being ellipsoidal be imposed, but not the condition of being a figure of revolution. Whatever be the moment of momentum, there is one, and only one revolutional figure of equilibrium, as we have seen in § 776; we now add: cal round an axis. (1) The equilibrium in the revolutional figure is stable, or Instability of oblate spheroid and stability of Jacobian (2) When the moment of momentum is less than that which figure. makes ƒ = 1·39457 (or eccentricity = 81266) for the revolutional figure, this figure is not only stable, but unique. (3) When the moment of momentum is greater than that which makes ƒ= 1.39457 for the revolutional figure, there is, besides the unstable revolutional figure, the Jacobian figure (§ 778 above) with three unequal axes, which is always stable if the condition of being ellipsoidal is imposed. But, as will be seen in (f) below, the Jacobian figure, without the constraint to ellipsoidal figure, is in some cases certainly unstable, though it seems probable that in other cases it is stable without any constraint. . Unstable Jacobian figures. (f) Looking back now to § 778 and choosing the case of a a great multiple of b, we see obviously that the excess of b above c must in this case be very small in comparison with c. Thus we have a very slender ellipsoid, long in the direction of a, and approximately a prolate figure of revolution relatively to this long a-axis, which, revolving with proper angular velocity round its shortest axis c, is a figure of equilibrium. The motion so constituted, which, without any constraint is, in virtue of § 778 a configuration of minimum energy or of maximum energy, for given moment of momentum, is a configuration of minimum energy for given moment of momentum, subject to the condition that the shape is constrainedly an ellipsoid. From this proposition, which is easily verified, in the light of § 778, it follows that, with the ellipsoidal constraint, the equilibrium is stable. The revolutional ellipsoid of equilibrium, with the same moment of momentum, is a very flat oblate spheroid; for it the energy is a minimax, because clearly it is the smallest energy that a revolutional ellipsoid with the same moment of momentum can have, but it is greater than the energy of the Jacobian figure with the same moment of momentum. (g) If the condition of being ellipsoidal is removed and the liquid left perfectly free, it is clear that the slender Jacobian ellipsoid of (ƒ) is not stable, because a deviation from ellipsoidal figure in the way of thinning it in the middle and thickening it towards its ends, would with the same moment of momentum give less energy. With so great a moment of momentum as to give an exceedingly slender Jacobian ellipsoid, it is clear that Configura- another possible figure of equilibrium is, two detached approximately spherical masses, rotating (as if parts of a solid) round an axis through their centre of inertia, and that this figure is stable. It is also clear that there may be an infinite number of such stable figures, with different proportions of the liquid in the two detached masses. With the same moment of momentum there are also configurations of equilibrium with the liquid in divers proportions in more than two detached approximately spherical masses. tion of two detached rotating masses stable. (h) No configuration in more than two detached masses, tion of two rotating has secular stability according to the definition of (k) below, Configuraand it is doubtful whether any of them, even if undisturbed by detached viscous influences, could have true kinetic stability: at all masses events, unless approaching to the case of the three material points proved stable by Gascheau (see Routh's "Rigid Dynamics," § 475, p. 381). (2) The transition from the stable kinetic equilibrium of a liquid mass in two equal or unequal portions, so far asunder that each is approximately spherical, but disturbed to slightly prolate figures (found by the well-known investigation of equilibrium tides, given in § 804 below), and to the more and more prolate figures which would result from subtraction of energy without change of moment of momentum, carried so far that the prolate figures, now not even approximately elliptic, cease to be stable, is peculiarly interesting. We have a most interesting gap between the unstable Jacobian ellipsoid when too slender for stability, and the case of smallest moment of momentum consistent with stability in two equal detached portions. The consideration of how to fill up this gap with intermediate figures, is a most attractive question, towards answering which we at present offer no contribution. (j) When the energy with given moment of momentum is either a minimum or a maximum, the kinetic equilibrium is clearly stable, if the liquid is perfectly inviscid. It seems probable that it is essentially unstable, when the energy is a minimax; but we do not know that this proposition has been ever proved. (k) If there be any viscosity, however slight, in the liquid, or if there be any imperfectly elastic solid, however small, floating on it or sunk within it, the equilibrium in any case of energy either a minimax or a maximum cannot be secularly stable : and the only secularly stable configurations are those in which the energy is a minimum with given moment of momentum. It is not known for certain whether with given moment of momentum there can be more than one secularly stable configuration of equilibrium of a viscous fluid, in one continuous mass, but it seems to us probable that there is only one. stable. Digression 779. A few words of explanation, and some graphic illustraharmonics. tions, of the character of spherical surface harmonics may pro on spherical spheroid. mote the clear understanding not only of the potential and hydrostatic applications of Laplace's analysis, which will occupy us presently, but of much more important applications to be made in Vol. II., when waves and vibrations in spherical fluid or elastic solid masses will be treated. To avoid circumloHarmonic cutions, we shall designate by the term harmonic spheroid, or spherical harmonic undulation, a surface whose radius to any point differs from that of a sphere by an infinitely small length varying as the value of a surface harmonic function of the position of this point on the spherical surface. The definitions of spherical solid and surface harmonics [App. B. (a), (b), (c)] show that the harmonic spheroid of the second order is a surface of the second degree subject only to the condition of being approximately spherical: that is to say, it may be any elliptic spheroid (or ellipsoid with approximately equal axes). Generally a harmonic spheroid of any order i exceeding 2 is a surface of algebraic degree i, subject to further restrictions than that of merely being approximately spherical. Let S be a surface harmonic of the order i with the coefficient of the leading term so chosen as to make the greatest maximum value of the function unity. Then if a be the radius of the mean sphere, and c the greatest deviation from it, the polar equation of a harmonic spheroid of order i will be if S is regarded as a function of polar angular co-ordinates, 0, p. Considering that c/a is infinitely small, we may reduce this to an equation in rectangular co-ordinates of degree i, thus:-Squaring each member of (1); and putting cri/a+ for c/a, from which it differs by an infinitely small quantity of the second order, we have Harmonic nodal cone and line. This, reduced to rectangular co-ordinates, is of algebraic degree i 780. The line of no deviation from the mean spherical surface is called the nodal line, or the nodes of the harmonic spheroid. It is the line in which the spherical surface is cut on spherical Harmonic by the harmonic nodal cone; a certain cone with vertex at the Digression for the equation of the harmonic nodal cone. As V is [App. Theorem B. (a)] a homogeneous function of degree i, we may write -1 V1 = H ̧1 + H‚ ̧≈→1 + H ̧12 + H ̧13 + etc.............(4), where H is a constant, and H,, H., H2, etc., denote integral homogeneous functions of x, y of degrees 1, 2, 3, etc.; and then the condition V1 = 0 [App. B. (a)] gives which express all the conditions binding on H,, H,, H2, etc. Now suppose the nodal cone to be autotomic, and, for brevity. and simplicity, take OZ along a line of intersection. Then z = a makes (3) the equation in x, y, of a curve lying in the tangent plane to the spherical surface at a double or multiple point of the nodal line, and touching both or all its branches in this point. The condition that the curve in the tangent plane may have a double or multiple point at the origin of its co-ordinates is, when (4) is put for V1, H1=0; and, for all values of x, y, H1 =0. 1 we have A + B = 0. This shows that the two branches cut one another at right angles. If the origin be a triple, or n-multiple point, we must have * "Summary of the Properties of certain Stream-Lines." Phil. Mag., Oct. 1864. regarding VOL. II. 22 |