From this result we see, that the period of the forced vibration is three times that of the force; accordingly the secondary waves will be of longer period, and consequently less refrangible, than the impinging waves.

Calorescence.

332. This phenomenon is the reverse of fluorescence, and consists in the conversion of waves of long period into waves of shorter period. Calorescence is well exhibited by the experiment of Tyndall already described under the head of spectrum analysis1, in which the light from an electric lamp is sifted of the luminous rays, by passing it through a solution of iodine in disulphide of carbon, which only allows the infra-red rays to pass through.

333. In order to obtain a mechanical model which will illustrate calorescence, we may revert to the differential equation (1). It can be verified by trial, that the complementary function is y= a cn (aut+a), k=2-1.... ...(2),

where a and a are the constants of integration. From this result it follows, that the amplitude of the free vibration is a, and its period is 4K/μa, which is inversely proportional to the amplitude. Hence the period diminishes as the amplitude increases.

Equation (2) may still be regarded as the complete solution of (1), provided we suppose that a and a, instead of being constants, are functions of the time, and their values might be found by the method of variation of parameters. If now, we suppose that the molecular forces are such, that a increases slowly with the time, we may illustrate the conversion of waves of dark heat into waves of light. When the waves of dark heat first fall on the substance and the forces begin to act, the amplitude is very small, and consequently the period is very large. On both these grounds therefore, the vibrations are incapable of affecting the senses. the forces continue to act, the amplitude increases, whilst the period diminishes, and the vibrations become sensible as heat; in other words the substance begins to get hot. As this process continues, the substance becomes red hot, and then intensely luminous. As the amplitudes cannot go on increasing indefinitely with the time, we must suppose that after the expiration of a certain period, the

1 Ante, p. 286.

As

B. O.

20

condition of the substance changes owing to liquefaction or vapourization, and that the equation by means of which the original state of things was represented, no longer holds good.

Phosphorescence.

334. When light is incident upon certain substances, such as the compounds of sulphur with barium, calcium or strontium, it is found that they continue to shine, after the light has been removed. This phenomena is called phosphorescence.

Phosphorescence is closely allied to fluorescence, inasmuch as it is usually produced by rays of high refrangibility, and the refrangibility of the phosphorescent light is generally less than that of the light by which it is produced. The principal distinction between the two phenomena is, that fluorescence lasts only as long as the exciting cause continues, whilst phosphorescence lasts some time after it has been removed.

335. In order to give a mechanical explanation of phosphorescence, we shall employ an acoustical analogue, which will frequently be made use of, and which will be fully worked out in § 337. Let plane waves of sound be incident upon a sphere, whose radius is small in comparison with the lengths of the waves of sound; and let the sphere be attached to a spring, so as to be capable of vibrating parallel to the direction of propagation of the waves. Then it is known', and will hereafter be proved, that the effect of the waves of sound will be to cause the sphere to vibrate. If the strength of the spring is such, that the force due to it is proportional to the displacement of the sphere, the forced period of the latter will be equal to that of the impinging waves of sound; if the law of force depends upon some power of the displacement, the forced period of the sphere will be different; but in either case secondary waves will be thrown off. These secondary waves will travel away into space carrying energy with them, which has been in the first instance communicated to the sphere by the incident waves, and then communicated back again to the air in the form of secondary waves. If the cause which produces the incident waves be removed, the sphere will still continue to vibrate, but it

1 Lord Rayleigh, Theory of Sound, Ch. XVII.

cannot go on vibrating indefinitely, because the energy which it possessed at the instant at which the incident waves were stopped, will gradually be used up in generating secondary waves, and will be carried away into space by them; hence the sphere will ultimately come to rest, and no more secondary waves will be produced.

Now although the molecules of a non-phosphorescent substance cannot be supposed to come to rest immediately the exciting cause is removed, yet the time during which they continue to be in motion is too short to be observed; but owing to the peculiar molecular structure of phosphorescent substances, the molecules remain in motion for a longer period. Hence a luminous glimmer exists for some time after the incident light has been cut off.

CHAPTER XVII.

THEORIES BASED ON THE MUTUAL REACTION BETWEEN ETHER AND MATTER.

336. IN the present Chapter, we shall give an account of some of the attempts which have been made to explain on dynamical grounds the phenomena described in the previous Chapter.

It may be regarded as an axiom, that when ethereal waves impinge upon a material substance, the molecules of the matter of which the substance is composed, are thrown into a state of vibration. This proposition is quite independent of any hypothesis, which may be made respecting the constitution of the ether, the molecular forces called into action by the displacements of the molecules of matter, or the forces arising from the action of ether upon matter. It may therefore be employed as the basis of a theory, in which the ether is regarded, either as a medium possessing the properties of an elastic solid, or as one which is capable of propagating electromagnetic disturbances as well as luminous waves. The difficulties of constructing theories of this description arise, not only from the fact that the properties of the ether are a question of speculation, but also because the forces due to the action of matter upon matter, and of ether upon matter, are unknown.

During the last five and twenty years, numerous attempts have been made by continental writers to develop theories of this description, and an account of them will be found in Glazebrook's Report on Optical Theories'. It cannot, I think, be said that any of these theories are entirely satisfactory; but at the same time,

1 Brit. Assoc. Rep. 1886.

VIBRATIONS OF A SPHERE IN AIR.

309

they clearly indicate the direction, in which we must look for an explanation of the phenomena, which they attempt to account for.

337. As an introduction to the subject, we shall work out and discuss the problem of the sphere vibrating under the action of plane waves of sound, which has been referred to in the last Chapter. The problem itself was first solved by Lord Rayleigh1, and has been reproduced by myself2 in an approximate form; but there are several additional points which require consideration.

Let c be the radius of the sphere; let = 2π/, where X is the wave-length; and let a be the velocity of sound. Then the velocity potential of the incident waves may be taken to be pekat, where p = . Now if μ = cos 0,

where P, is the zonal harmonic of degree n, and3

If deixat be the velocity potential of the secondary waves, we may assume1

$ = ΣAn¥nPn

and f is the function defined by (40) of § 231.

If X be the resistance due to the pressure of the air,

X = − 2πрc2 ["*" (4′ + 4) ixαeTMat cos ◊ sin Ode

where M' is the mass of the fluid displaced.

..(2),

If Veat be the velocity of the sphere, the boundary condition

Stokes, On the communication of vibrations from a vibrating body to the

atmosphere. Phil. Trans. 1868.