We shall now suppose, that the radius of the sphere is so small in comparison with the wave-length, that the fourth and higher powers of xc may be neglected. Hence if & be the displacement of the sphere, = − } M′ [ {(2 + x2c2) § + Kac3α§} − {ıκα (2 + x2c2) Substituting the value of F, from (1), and remembering that Kα = 2π/T, where is the period of the waves of sound, we finally obtain from (2) and (3), If the force due to the spring is proportional to the displacement of the sphere, it follows that this force is equal to 4π M§/T', is the free period of the sphere; whence the equation of motion is where To integrate this equation, assume = Ae2t, then 338. From these results we draw the following conclusions. Equation (4), which is the equation of motion, contains a viscous term, that is a term proportional to the velocity. This term arises from the circumstance, that the sphere is continually losing the energy which it receives from incident waves, by generating secondary waves, which travel away into space carrying energy with them. If therefore the supply of energy be stopped, OPTICAL APPLICATIONS. 311 by removing the cause which produces the impinging waves, the sphere will gradually get rid of all its energy, and will ultimately come to rest. The time which elapses before the sphere comes to rest, will depend upon the value of the modulus of decay; if this quantity is small, the vibrations will die away almost instantaneously; but if the modulus of decay is larger, the vibrations will continue for a sufficient time to enable our senses to take cognizance of them. 339. Now whatever supposition we make concerning the mutual reaction between ether and matter, it is practically certain that the motion of a molecule of matter will be represented by an equation, whose leading features are the same as (4), although the equation itself may be of far greater complexity. We therefore infer, that the molecular structure of non-phosphorescent substances is such, that the modulus of decay is so small as to be inappreciable; whilst the molecular structure of phosphorescent substances is such, that the modulus of decay is considerably larger. 340. We must now consider the amplitude of the vibrations of the sphere, which is given by (5). The density of all gases is exceedingly small, compared with the densities of substances in the solid or liquid state; consequently M' is very small compared with M. Hence the amplitude of the sphere is exceedingly small in comparison with that of the incident waves (which has been taken as unity), unless 7 and 7' are nearly equal. When 7=7', the large term in the denominator disappears, and A is approximately equal to 3/a. Under these circumstances, the amount of energy communicated by the incident waves to the sphere, is very much greater than what it would have been, if the difference between 7 and 7' were considerable. T 341. Let us now consider a medium, such as a stratum of sodium vapour. We may conceive the molecules of the medium to be represented by a very large number of small spheres, and the molecular forces to be represented by springs. The medium will therefore have one or more free periods of vibration. The interstices between the molecules are filled with ether, which is represented by the atmosphere. When waves of light pass through the medium, the molecules will be set into vibration, and a certain amount of energy will be absorbed by them; but since the mass of a molecule is exceedingly large compared with the mass of the ether which it displaces, the amplitudes of the vibrations of the molecules will be very small, and very little energy will be absorbed, unless the period of the waves is equal, or nearly so, to one of the free periods of the system. But in the case of equality of the free and forced periods, the amplitudes will be so large, that a great deal of energy will be taken up by the molecules, and of the energy which entered the stratum of vapour, very little will emerge. Light will therefore be absorbed. The absorption bands produced by sodium vapour may therefore be explained, by supposing that sodium vapour has two free periods, which are very nearly equal to one another, and accordingly produce the double line D in its absorption spectrum. Hydrogen, on the other hand, has three principal free periods, which are separated from one another by considerable intervals. The occurrence of an imaginary term in the denominator of A shows, that A can never become infinite for any real value of 7. This remark will be found of importance later on. Lord Kelvin's Molecular Theory. 342. We shall now consider a theory, which was developed by Sir W. Thomson, now Lord Kelvin, in his lectures on Molecular Dynamics, delivered at Baltimore in 1884. The molecules of matter are represented by a number of hollow spherical shells, connected together by zig-zag massless springs; and the outermost shell is connected by springs to a massless spherical envelop, which is rigidly connected with the ether. The space between any two shells is supposed to be a vacuum, and transparent and other substances are supposed to consist of a great number of such shells, which may be imagined to represent the molecules. The degree of complexity of the molecule will depend upon the number of shells which it contains; and we can by this means represent chemical compounds of every degree of complexity. We shall first of all investigate the motion of a single molecule, on the supposition that the centres of all the spherical shells are vibrating along a fixed straight line. We shall also suppose, that the force exerted by the springs joining two consecutive shells is proportional to their relative displacements; and that the force LORD KELVIN'S MOLECULAR THEORY. 313 - x1; may be exerted by the ether on the envelop is proportional to where a is the displacement of the outermost shell, and regarded indifferently, either as the displacement of the envelop, or of the ether in contact with it. 343. Let m;/472 be the mass of the ith shell, x; its displacement, C; the strength of the spring connecting the ith and (i-1)th shells. Then the equations of motion of the system of shells will be the equations of motion will reduce to the form - Cixi-1 = α¿Xi + Ci+1 Xi+1 (6). (7), (8). (9), so that we may regard, as equal to the displacement of the ether. If we suppose that the jth shell is attached to a fixed point, the jth equation of motion will be Although it is scarcely admissible to suppose, that the jth shell is attached to a fixed point, yet if we suppose that the (j+1)th shell consists of a solid nucleus, whose mass is large compared with that of the other shells, its motion will be sufficiently small to be neglected. = There are j-2 equations of the form (8), which are obtained by putting i 2, 3, ... j - 1; and these equations together with (9) and (10) furnish altogether j equations, from which the j-1 quantities a, ... ; can be eliminated, and we shall thus obtain a relation between x, and Also since j+1=0; j+1=∞, and therefore Uj = αj...... From these equations we see, that u can be expressed in the form of the continued fraction Putting for a moment & for d/dr-2, it follows from (7) that From this equation we see, that du/dr is always negative; and therefore u; diminishes as 7 increases. |