absolute physical constants of molecular theory, such as the
constant of Avogadro, which are in close accord with other
recent determinations. But on the other hand these determinations
are already involved in the earlier formula of Rayleigh,
which expresses the distribution for long waves, based merely
on the Maxwell-Boltzmann principle of the equable partition
of the energy among the high free periods belonging to the
enclosure which contains it. It is maintained by leans that the
reason why this principle is of avail only for very long wavelengths
is that a steady state is never reached for the shorter ones,
a doctrine which as he admits would entirely remove the foundations
of the application of thermodynamic principles to this
subject. By an argument based on the theory of dimensions,
Lorentz has been led to the conclusion that consistency between
temperatures, as measured molecularly, and as measured by the
laws of radiation, requires that the ultimate indivisible electric
charges or electrons must be the same in all kinds of matter.
The abstract statistical theory of entropy, which is here invoked, admits of generalization in a way which is a modification of that of Planck, itself essentially different from the earlier idea of Boltzmann. The molecules of matter, whose interactions control physical phenomena, including radiation, are too numerous to be attended to separately in our knowledge. They, and the phenomena in which they interact, must thus be sorted out into differential groups or classes. Elements of energy of specified types might at first sight constitute such classes: but the identity of a portion of energy cannot be traced during its transformations, while an element of physical disturbance can be definitely followed, though its energy changes by interaction with other elements as it proceeds. The whole disturbance may thus be divided into classes, or groups of similar elements, each with permanent existence: and these may be considered as distributed in series of cells, all equivalent in extent, which constitute and map out the material system or other domain of the phenomena. The test of this equivalence of extent is superposition, in the sense that the same element of disturbance always occupies during its wanderings the same number of cells. This framework being granted, the probability of any assigned statistical distribution of the elements of disturbance now admits of calculation; and it represents, as above, the logarithm of the entropy of that distribution, multiplied however by a coefficient which must depend on the minuteness of scale of the statistics. But in the calculation, all the physical laws which impose restrictions on the migrations of the elements of disturbance must be taken into account; it is only after this is done that the rest of the circumstances can be treated as fortuitous. All these physical laws are, however, required and used up in determining the complex of equivalent cells into which the system which forms the seat of the energy is mapped out. On this basis thermodynamics can be constructed in a priori abstract fashion, and with deeper and more complete implications than the formal Carnot principle of negation of perpetual motions can by itself attain to. But the ratio of the magnitude of the standard element of disturbance to the extent of the standard cell remains inherent in the results, appearing as an absolute physical constant whose value is determined somehow by the other fundamental physical constants of nature. A prescribed ratio of this kind is, however, a different thing from the hypothesis that energy is constituted atomically, which underlies, as Lorentz pointed out, Planck's form of the theory. It has indeed already been remarked that the mere fact of the existence of a wave-length λm of maximum radiation, whether obeying Wien's law λmT=constant or not, implies by itself some, prescribed absolute physical quantity of this kind, whose existence thus cannot be evaded, though we may be at a loss to specify its nature.
13. Modification by a Magnetic Field.—The theory of exchanges of radiation, which makes the equilibrium of radiating bodies depend on temperature alone, requires that, when an element of surface of one body is radiating to an element of surface of another body at the same temperature, the amounts of energy interchanged (when reflexion is counted in along with radiation) should be equal. This proposition is a general dynamical consequence—on the basis of the laws of reciprocity developed in this connexion (after W. Rowan Hamilton) mainly by Helmholtz, Kirchhoff, and Rayleigh—of the form of the equations of propagation of vibrations in the medium. But in a material medium under the influence of a strong magnetic field these equations are altered by the addition of extraneous terms involving differential coefficients of the third order, and the dynamical consistency of the cardinal principle of the theory of exchanges is no longer thus directly verified. A system of this kind has, in fact, been imagined by Wien in which the principle is imperfectly fulfilled. A beam coming from a body A, and polarized by passage through a nicol, may have its plane of vibration rotated through half a right angle by crossing a magnetically active plate, and may then pass through another nicol, properly orientated for transmission, so as finally to fall on another body B. On the other hand, the radiation from B which gets through this adjacent nicol will have its plane of vibration rotated through another half right angle by the magnetically active plate, and so will not get through the first nicol to the body A. Such possibilities of unequal exchange of radiation between A and B are the result of the want of reversibility of the radiation in the extraneous magnetic field, which might have been expected to lead to proportionate inequalities of concentration; in this example, however, though the defect of reversibility is itself slight, its results appear at first sight to prevent any equilibrium at all. But a closer examination removes this discrepancy. In order to make the system self-contained, reflectors must be added to' it, so as to send back into the sources the polarized constituents that are turned aside out of the direct line by the nicols. Then, as Brillouin has pointed out, and as in fact Rayleigh had explained some years before, the radiation from B does ultimately get across to A after passage backward and forward to the reflectors and between the nicols: this, it is true, increases the length of its path, and therefore diminishes the concentration of a single narrow beam, but any large change of path would make the beam too wide for the nicols, and thus require other corrections which may be supposed to compensate. The explanation of the slight difference that is to be anticipated on theoretical grounds might conceivably be that in such a case the magnetic influence, being operative on the phases, alters the statistical constitution of the radiation of given wave-length from the special type that is in equilibrium with a definite temperature, so that after passage through the magnetic medium it is not in a condition to be entirely absorbed at that temperature; there would then be some other element, in addition to temperature, involved in equilibrium in a magnetic field. If this is not so, there must be some thermodynamic compensation involving reaction, extremely small, however, on the magnetizing system.
14. Origin of Spectra.—In addition to the thermal radiations of material substances, those, namely, which establish temperature-equilibrium of the enclosure in which they are confined, there are the fluorescent and other radiations excited by extraneous causes, radiant or electric or chemical. Such radiations are an indication, by the presence of higher wave-lengths than belong in any sensible degree to the temperature, that the steady state has not arrived; they thus fade away, either immediately on the cessation of the exciting cause, or after an interval. The radiations, consisting of definite narrow bright bands in the spectrum, that are characteristic of the gaseous state in which each molecule can vibrate freely by itself, are usually excited by electric or chemical agency; thus there is no ground for assuming that they always constitute true temperature radiation. The absorption of these radiations by strata of the same gases at low temperatures seems to prove that the unaltered molecules themselves possess these free periods, which do not, therefore, belong specially to dissociated ions. Although very difficult to excite directly, these free vibrations are then excited and absorb the energy of the incident waves, under the influence of resonance, which naturally becomes extremely powerful when the tuning is exact; this
