We have still to express the condition that (47) should be the equation of a bounding surface. This condition [(10) Art. 10,] becomes in the present case Eliminating S between (48) and (49) we find The motion consists therefore in general of a series of superposed oscillations, the periods τ of which are obtained by putting n = 1, 2, 3, &c., in (50). The longest period is that for which n = 1, in which case (T, infinitely small) is the centre is near the origin. as we should expect. In fact r = b + T1............... .(51). .(52). equation of a sphere of radius b whose The fluid and solid are then equivalent to a single spherical mass of uniform density, so that there is always equilibrium when the surface is of the form (52). If p>σ, 7, is imaginary, the value of 7,2 given by (51) being then negative. This indicates that the equilibrium of the ocean, when in the form of a sphere concentric with the earth, is unstable. The ocean would in fact, if disturbed, tend to heap itself up on one side *. * See Thomson and Tait, Nat. Phil. Art. 816. If a = 0, we get the case treated by Thomson, viz. the case in which a mass of liquid oscillates about the spherical form under the mutual gravitation of its parts. The formula (50) becomes 42b 2n+1 2 "It is worthy of remark that the period of vibration thus calculated is the same for the same density of liquid, whatever be the dimensions of the globe. "For the case of n = 2, or an ellipsoidal deformation, the length of the isochronous simple pendulum becomes b, or one and a quarter times the earth's radius, for a homogeneous liquid globe of the same mass and diameter as the earth; and therefore for this case, or for any homogeneous liquid globe of about 5 times the density of water, the half-period is 47m 12s*" "A steel globe of the same dimensions, without mutual gravitation of its parts, could scarcely oscillate so rapidly, since the velocity of plane waves of distortion in steel is only about 10,140 feet per second, at which rate a space equal to the earth's diameter would not be travelled in less than 1h 8m 40s+." If on the other hand the depth h of the ocean be small compared with the radius of the earth, we have, writing in (50) b = a+h, and For large values of n the distance from crest to crest in the surface represented by (47) is small compared with the radius of the earth. The propagation of the disturbance then takes place according to the laws investigated in Art. 147. The formula (53) then becomes, approximately, T = 2πα÷n√gh, a result which the student who is familiar with the properties of spherical harmonics will easily see to be consistent with (7). * Phil. Trans. 1863, p. 610. + Phil. Trans. 1863, p. 573. + Mécanique Celeste, Livre 4me, Art. 1. CHAPTER VIII. WAVES IN AIR *. 165. We investigate in this chapter the general laws of the propagation of small disturbances produced in a mass of air (or other gas) originally at rest at a uniform temperature, avoiding such details as are more properly treated in books specially devoted to the theory of Sound. Plane waves. We take first the case where the motion is in one dimension x only. Let be the displacement at time t of the particles which in the undisturbed state occupy the position x. The stratum of air originally bounded by the planes x and x + dx is at the time t bounded by the planes x + §, and x + §+ (1 + d) da, so that the equation of continuity is Ро dx ..(1), where p is the density in the undisturbed state. The equation of motion of the stratum is and if we suppose the condensations and rarefactions to succeed one another so rapidly that there is no sensible gain or loss of heat in any stratum by conduction or radiation, the relation between p and p is ρ p = k' p2....... .(3). * This chapter was written independently of the corresponding portion of Lord Rayleigh's Theory of Sound, the second volume of which did not come into the author's hands until after the MS. of this treatise had been despatched to England (October 1878). 166. Let us denote by s the 'condensation,' i.e. the value of P-Po at any point. If this be small, we have, by (1), Ро nearly; and if as in all ordinary cases of sound the condensation, and its rate of variation from point to point, be both small quantities whose squares, &c., may be neglected, the equation (4) be which denotes two systems of waves travelling with velocity c, one in the positive, the other in the negative direction of x. Comparing (5) with the ordinary expression of the laws of Boyle and Charles, viz. we find p = kp (1 + a0), c2 = ky(1 + a0), so that the velocity of propagation c depends, for the same gas, only on the temperature 0. For a wave travelling in one direction only, say that of x positive, we have and therefore u = CS..... ...... (8). For a wave travelling in the direction of a negative we should have It will be noticed that there is an exact correspondence between the above approximate theory, and that of 'long' waves in water (Art. 147). By the substitution of the words 'condensation of air' for 'elevation of water,' and 'rarefaction' for 'depression,' the two questions become identical. The analogy becomes however less close when we proceed to a higher degree of approximation. Spherical waves. 167. Let us suppose that the disturbance is symmetrical with respect to a fixed point, which we take as origin. The motion is then necessarily irrotational, so that a velocity-potential & exists, which is a function of r, the distance of any point from the origin, and t, only. If as before we neglect the squares of small quantities, we have Now [dp__dd dt γ whence, writing p=p。(1+s), and neglecting the square of s, we find To form the equation of continuity we remark that, owing to the difference of flux across its inner and outer surfaces, the space bounded by the spheres r and r + dr is gaining matter at the rate |