# 1911 Encyclopædia Britannica/Radiation, Theory of

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​RADIATION, THEORY OF. The physical activities that flourish on the surface of the earth derive their energy, in a form which is highly available thermodynamically, from the radiation of the sun. This has been ascertained to be dynamic energy, transmitted in waves by the vibrations of a medium occupying space, as the energy of sound is transmitted by the vibrations of the atmosphere. The elasticity that transmits it may be assumed to be mathematically perfect: any slight loss in transit of the light from the most distant stars, which recent statistical comparisons of brightness with distance may possibly indicate, is to be explained far more suitably by the presence of nebulous matter than by any imperfection of the aether. The latter would thus be the one perfect frictionless medium known to us: it could not be such if it were constituted, like matter, of independent molecules. It is thus on a higher plane, and may even be considered to be »a dynamical' specification of space itself. A molecule of matter is a kinetic system compounded of simpler elements; its energy may be classified into constitutive energy essential to its continued existence, and vibratory energy which it can receive from or radiate away into aether. A piece of matter isolated in free aether would in time lose ag energy of the latter type by radiation; but the former will remain so long as the matter persists, along with the energy of the uniform translator motion to which it is ultimately reduced. Thus all matter is in continual exchange of vibratory energy with the aether: it is with the laws of this exchange of energy that the general theory of Radiation deals, as distinguished from the mechanism of the ethereal vibrations, which is usually treated as the Theory of Light (see Aether).
1. The foundation of this subject is the principle, arrived ​ at independently by Balfour Stewart and Kirchhoff about the year 1858, that the constitution (§ 6) of the radiation which pervades an enclosure, surrounded by bodies in a steady thermal state, must be a function of the temperature of those bodies, and of nothing else. It was subsequently pointed out by Stewart (Brit. Assoc. Report, 1871) that if the enclosure contains a radiating and absorbing body which is put in motion, all being at the same temperature, the constituents of the radiation in front of it and behind it will differ in period on account of the Doppler-Fizeau effect, so that there will be an opportunity of gaining mechanical work in its settling down to an equilibrium; there must thus be some kind of thermodynamic compensation, which might arise either from ethereal friction, or from work required to produce the motion of the body against pressure exerted on it by the surrounding radiation. The hypothesis of friction is now excluded in ultimate molecular physics, while the thermodynamic bearing of a pressure exerted by radiation, such as is demanded by Maxwell's electric theory, has been more recently developed on other lines by Bartoli and Boltzmann (1884), and combined with that of the Doppler effect by W. Wien (1893) in development of the ideas above expressed.
The original reasoning of Stewart and Kirchhoff rests on the dynamical principle, that by no process of ordinary reflexion or transmission can the period, and therefore the wave-length, of any harmonic constituent of the radiation be changed; each constituent remains of the same wave-length from the time it is emitted until the time it is again absorbed. If we imagine a field of radiation to be enclosed within perfectly reflecting walls, then, provided there is no material substance in the field which can radiate and absorb, the constitution of the radiation in it may be any whatever, and it will remain permanent. It is only the presence of material bodies that by their continued emission and absorption can transform the surrounding radiation towards the unique constitution which corresponds to their temperature. We can define the temperature of an isolated field of radiation, of this definite ultimate constitution, to be the same as that of the material bodies with which it would thus be in equilibrium. Further, the mutual independence of the various constituents of any field of radiation enclosed by perfect reflectors allows us to assign a temperature to each constituent, such as the part involving wave-lengths lying between λ and λ+δλ; that will be the temperature of a material system with which this constituent by itself is in equilibrium of emission and absorption, But to reason about the temperature of radiation in this way we must be sure that it completely pervades the space, and has no special direction; this is ensured by the continual reflexions from the walls of the enclosure. The question of the temperature of a directed wave-train travelling through space, such as a beam of light, will come up later. The temperature of each constituent in a region of undirected radiation is thus a function of its wave-length and its intensity alone. It is the fundamental principle of thermodynamics, that temperatures tend to become uniform. In the present case of a field of radiation, this equalization cannot take place directly between the various constituents of the radiation that occupy the same space, but only through the intervention of the emission and absorption of material bodies; the constituent radiations are virtually partitioned off adiabatically from direct interchange. Thus in discussing the transformations of temperatures of the constituent elements of radiation, we are really reasoning about the activity of material bodies that are in thermal equilibrium with those constituents; and the theoretical basis of the idea of temperature, as depending on the fortuitous residue of the energy of molecular motions, is preserved.
2. Mechanical Pressure of Undulatory Motions.—Consider a wave-train of any kind, in which the displacement is ξ=a cos m(x+ct) so that it is propagated in the direction in which x decreases; let it be directly incident on a perfect reflector travelling towards it with velocity v, whose position is therefore given at time t by x=vt. There will be a reflected train given by ξ′=a′ cos m′(x−ct), the velocity of propagation c being of course the same for both. The disturbance does not travel into the reflector, and must therefore be annulled at its surface; thus when x=vt we must have ξ+ξ′=0 identically. This gives a′= −a, and m′(c−v)=m(c+v). The amplitude of the reflected disturbance is therefore equal to that of the incident one; while the wave-length is altered on the ratio gli, which is approximately I-Zig, where v/c is small, and is thus in agreement with the usual statement of the Doppler effect. The energy in' the wave-train being half potential and half kinetic, it is given by the integration of p(d§ /dt)2 along the train, where p represents density. In the reflected train it is therefore augmented, when equal lengths are compared, in the 2
fatlo ; but the length of the train is diminished by the reflexion in the ratio git; hence on the whole the energy transmitted per unit time is increased by the reflexion in the ratio 3. This increase per unit time can arise only a-v ~
from work done by the advancing reflector against pressure exerted by the radiation. That pressure, per unit surface, must therefore be equal to the fraction £1 of the energy in a length 2 2
c-I-v of the incident wave-train; thus it is the fraction 251% of the total density of energy in front of the reflector, belonging to both the incident and reflected trains. When v is small compared with c, this makes the pressure equal to the density of vibrational energy, in accordance with Maxwell's electrodynamic formula (Elec. and Mag., 1871).
The argument may be illustrated by the transverse vibrations of a tense cord, the reflector being then a lamina through a small aperture in which the cord passes; the lamina can thus slide along the cord and sweep the vibratory motion in front of it. In this case the force acting on the lamina is the resultant of the tensions T of the cord on the two sides of the aperture, giving a lengthwise force é-Td(E+E')2/dx” when, as usual, powers higher than the second of the ratio of amplitude to wave-length are neglected; this, when u/c is small, is an oscillatory force of amount 2p(dE/dt)2, whose time-average agrees with the value above obtained. If we consider a finite train of waves thus sent back from a moving reflector, the time integral of the pressure must represent force transmitted along the cord, or a gain of longitudinal momentum in the reflected waves, or both together.
When it is a case of transverse waves in an elastic medium, reflected by an advancing obstacle, the origin of the working pressure is not so obvious, because we cannot easily formulate a mechanism for the advancing reflector like that of the lamina above employed. In the case of light-waves we can, however, imagine an ideal material body, constituted of very small molecules, that would sweep them in front of .it with the same perfection as a metallic mirror actually reflects the longer Hertzian waves. The pressure will then be identified physically, as in the case of the latter waves, with the mechanical forces acting on the screening oscillatory electric current-sheet which is induced on the surface of the reflector. The displacement represented above by E, which is annulled at the reflector, may then be taken to be either the tangential electric force or the normal component of the vector whose velocity is the magnetic force. The latter interpretation is theoretically interesting, because that vector, which is the dynamical displacement in electron theory, usually occurs only through its velocity. The general case of oblique incidence can be treated on similar lines; each filament of radiation (ray) in fact exerts its own longitudinal push equal to its energy per unit length, and it is only a matter of summation.
The usual formula for the pressure of electric radiation is ​ derived from a theory, namely, that of the ordinary electrodynamic equations, which considers the velocity of the matter, or rather of the electrons associated with it, to be so small compared with that of radiation that the square of the ratio of these velocities can be neglected. The formula above obtained is of general application, and shows that for high values of v the pressure must fall off. It has been urged as an objection to the thermodynamic reversibility of a ray (§ 8) that the work of the radiant pressure exerted at its front is lost, as there is no obstacle to sustain it; but on an obstacle moving with the velocity of the wave-front the pressure would vanish, so that this objection does not now hold.
In every such case of an advancing perfect reflector the aggregate amplitude of the superposed incident and reflected wave-trains, of different wave-lengths and periods, will be represented by
mv C2 mc
E-{-5 -2a s1nc v(x - vt) sin6 U(x-ul),
thus the appearance presented will be that of a train of waves each of length (1-u/c)21r/m, and progressing with the velocity u of the reflector, which travels at one of the nodes of the train. This slowly travelling wave-train corresponds to the stationary train which would be produced by a stationary perfect reflector; but the amplitude is now a varying quantity which, once uniform vibration has been fully established along any path, may itself be described as running on after the manner of a superposed wave-train of very great wave-length (0/u-r)21r/m and of very great velocity 6”/v. A somewhat similar state of things arises when a wave-train is incident on a stationary reflector very nearly normally, as may sometimes be seen with incoming rollers along a shelving beach; the visible disturbance at a reflecting ridge, arising from each single wave-crest, then rushes along the ridge at a speed which is at first sight surprising, as it is enormously in excess of the speed possible for any simple train of waves travelling into quiescent aether. 3. Wien's Law.-Let us consider a spherical enclosure filled with radiation, and having walls of ideal perfectly reflecting quality so that none of the radiation can escape. If there is no material body inside it, any arbitrarily assigned constitution of this radiation will be permanent. Let us suppose that the radius a of the enclosure is shrinking with extremely small velocity u. A ray inside it, incident at angle L, will always be incident on the walls in its successive reflexions at the same angle, except as regards a negligible change due to the motion of the reflector (§ 2); and the length of its path between successive reflexions is 20 cos L. Each undulation on this ray will thus undergo reflexion at intervals of time equal to za cos L/0, where c is the velocity of light, and it is easily verified that on each reflexion it, is shortened by the fraction 2U cos L/c of itself: thus in the very long time T required to complete the shrinkage it is shortened by the fraction uTa, which is 5a/a where 5a is the total shrinkage in radius, and is independent of the value of L. The wave-length of each undulation in the radiation inside the enclosure is therefore reduced in the same ratio as the radius. Now suppose that the constitution of the enclosed radiation corresponded initially to a definite temperature. During the shrinkage thermal equilibrium must be maintained among its constituents; otherwise there would be a running down of their energies towards uniformity'of temperature, if material radiating bodies are present, which would be superposed on the mechanical operations belonging to the shrinkage, and the process could not be reversible. Such a state of affairs is not possible, for it would land us in processes of the following type. Expand the enclosure, gaining the mechanical work of the radiant pressure against its walls, whatever that may be. Then equalize the intensities of the constituent radiations to those corresponding to a common temperature, by taking advantage of the absorptions of material bodies at the actual temperatures of these radiations; when this is done, as it may actually be to some extent by aid of the sifting produced by partitions which transmit some kinds of radiation more rapidly than others, a further gain of work can be obtained at the expense of the radiant energy. Then contract the remaining radiant energy to its previous volume, which requires an expenditure of less work on the walls of the enclosure than the expansion of the greater amount of radiation originally afiorded; and, finally, gain still more work by again equalizing the temperatures of its constituents, The energy now remaining, being of smaller amount and under similar conditions, must have a temperature lower than the initial one. This process might be repeated indefinitely, and would constitute an engine without an extraneous refrigerator, violating Carnot's principle by deriving an unlimited supply of. mechanical work from thermal sources at a uniform temperature.
Thus, independently of any knowledge of the intensity of the mechanical pressure of radiation, or indeed of whether such a pressure exists at all, it is established that the shrinkage of the enclosure must directly transform the contained radiation to the constitution which corresponds to some definite new temperature. Now we have seen that the wave-lengths of its constituents are all reduced in the same ratio by this process. If, then, we can prove that the intensities of these constituents are also all changed in a common ratio by the reflexions at the shrinking envelope, it will follow that the distributions of the radiation among the various wave-lengths are, at these two temperatures, and therefore at any two temperatures, homologous, in the sense that the intensity curves, after the Wave-lengths in one of them have been reduced in a ratio depending definitely on the two temperatures, differ only in the absolute scale of magnitude of the ordinates.
This procedure modifies Wienfs argument by employing a uniformly shrinking spherical enclosure (cf. Brit. Assoc. Report, tooo). If the enclosure is not spherical, the angles of incidence at successive reflexions of the same ray will differ by finite amounts; we must then estimate the average effect of the shrinkage. In the form of enclosure here employed all rays are affected alike, and no averaging is required; while by the principle of Stewart and Kirchhoff what is established for any one form is of general validity.
4. Pressure of N afural Radiation.-The question reserved above has now to be settled. At first sight it might have appeared that the reflexion is simply total; but, as has been seen in § 2, the advancing perfect reflector does work against the pressure of the radiation, and this work must be changed into radiant energy and thus go to increase the intensity of the reflected ray. Considering electric radiation incident at angle L, the tangential electric force is annulled at the reflector; hence the amplitude of the electric vibration is conserved on reflexion, though its phase is reversed. As already seen, the wave-length is shortened approximately by the fraction 2U cos L/c in each reflexion; thus, just as in § 2, the energy transmitted per unit time per unit area is increased in the same ratio; andrallowing for the factor cos L of foreshortening, there is therefore a radiant pressure equal to 'the total density of radiant energy in front of the reflector multiplied by COS2L. This argument, being independent of the wave-length, applies to each constituent of the radiation in this direction separately; thus their energies are all increased in the same ratio by the reflexion, as was to be proved. When we are dealing with the natural radiation in an enclosure, which is distributed equally in all directions, this factor .COSZL must be averaged; and we thus attain Boltzmann's result that the radiant pressure is then one-third of the density of radiant energy in front of the reflector, this statement holding good as regards each constituent of the natural radiation taken separately.
5. Adiabaf-ic Relations.-Consider the enclosure filled with radiation of energy-density E at volume V, of any given constitution but devoid of special direction, and let it be shrunk to volume V - 5V against its own pressure; if the density thereby become E 1- 61-E, the conservation of the energy requires EV-I-§ E5V=(E-5E)(V-6V),
so, that § E<§ V+V5E=o, or E varies as V'ii.
Again-but now with a restriction to radiation with its energy ​ distributed as regards wave-length so as to be of uniform temperature-the performance of this mechanical work 1/3EδV has changed the energy of radiation EV from the state that is in equilibrium of absorption and emission with a thermal source at temperature T to the state in equilibrium with an absorber of some other temperature T−δT, and that in a reversible manner; thus by Carnot's principle
1/3EδV/EV＝−δT/T,
so that T varies as V$^{−1/3}$, or inversely as the linear dimensions when the enclosure is shrunk uniformly.
Combining these results, it appears that E varies as T$^4$; this is Stefan's empirical law for the complete radiation corresponding to the temperature, first established on these lines by Boltzmann. Starting from the principle that this radiation must be a function of the temperature alone, this adiabatic process has in fact given us the form of the function. These results cannot, however, be extended without modification to each separate constituent of the complete radiation, because the shrinkage of the enclosure alters its wave-length and so transforms it into a different constituent.
6. Law of Distribution of Energy.—The effect of compressing the complete radiation is thus to change it to the constitution belonging to a certain higher temperature, by shortening all its wave-lengths by the proportion of one-third of the compression by volume, the temperature being in fact raised by the same proportion; at the same time increasing in a uniform ratio the amounts corresponding to each interval δλ, so as to get the correct total amount of energy for the new temperature. In the compression each constituent alters so that Tλ) remains constant, and the energy E$_λ$δλ in the range oh in other respects changes as a function of T alone. Hence generally E$_λ$δλ must be of form F(T)f(Tλ)δλ. But for each temperature $\int _{0}^{\infty }$E$_λ$δλ is equal to E and so variers, as T$^4$, by Stefan's law; that is,
T$^{−1}$F(T)$\int _{0}^{\infty }$f(Tλ)d(Tλ) ∝ T$^4$
so that T$^{−1}$F (T) ∝ T$^4$. Thus, finally, E$_λ$δλ is of form ATQ/(T})5) or A)'°qb(T))5), which is Wien's general formula.
7. Transformation of a Single Constituent.—It is of interest to follow out this adiabatic process for each separate constituent of the radiation, as a verification, and also in order to ascertain whether anything new is thereby gained. To this end let now E, T)5) represent the intensity of the radiation between) and X-l-BX which corresponds to the temperature T. The pressure of this radiation, when it is without special direction, is in intensity one-third of this; thus the application of Carnot's principle shows, as before, that in adiabatic compression TocV'5, so that' a small linear shrinkage in the ratio 1-x raises T in the ratio 1-1-x. We have still to express the equation of energy. The vibratory energy E, T)6} . V in volume V, together with the mechanical work § E, T)67 . 3xV, yields the vibratory energy
E(r - x), T(r+x)}5}(1 - x) . V(r - 3x); J thus, writing E and E). or E, T) we have, neglecting xz, E(r+x) = (E-x>.%+xT§§)(1 -ix),
an dE
so that 5E+)-(KJFH, -o,
a partial differential equation of which the integral is E = A)'5qS (Th),
the same formula as was before obtained.
This method, treating each constituent of the radiation separately, has in one respect some advantage, in that it is necessary only to postulate an enclosure which totally reflects that constituent, this being a more restricted hypothesis than an absolutely complete reliector.
To determine theoretically the form of the function QS we must have some means of transforming one type of radiation »into another, different in essence from the adiabatic compression already utilized. The condition that the entropy of the independent radiations in an enclosure 'is a minimum when they are all transformed to the same temperature with total energy unaltered, is already implicitly fulfilled; it would thus appear that any further advance must involve (§ I 1) the dynamics of the radiation and absorption of material bodies. 8. Temperature of an Isolated Roy.-The temperature of each independent constituent of a radiation has here been taken to be a function of the intensity Ei, Where E, } is the energy per unit Volume in the range between wave-lengths X a.nd }-I-6}; the condition is, however, imposed' that this radiation is indifferent as to direction. When a beam of radiation 'travels without loss in a definite direction across a medium, its form varies as it progresses; but it is reversible inasmuch' as it can be turned back at any stage, or concentrated without loss, by perfect reflectors. If the energy of the beam has a temperature, its Value must therefore remain constant throughout the progress of the .beam, by the principle of Carnot. Now by virtue of a relation in geometrical optics, which on a corpuscular theory would be one aspect of the fundamental dynamical principle of Action, the cross-section, 65 at any place on the beam, and the conical angle 60: within which the directions of its rays are there included, are such that the value of V”26S5co is conserved along the beam, V being the velocity of propagation of the undulations. If we represent the amount of radiant energy transmitted per unit time across the section 6S of the beam by IδSδω, it will follow that in passing along the beam its intensity of illumination I varies as V$^{−2}$, or as the square of the index of refraction, provided there is no loss of energy in transmission. This condition requires that changes of index shall be gradual, otherwise there would be loss of energy by partial reflexions; in free aether I is itself constant along the beam. The volume-density of the energy in any part of the directed beam is V$^{−1}$Iδω; it is thus inversely as the solid angular concentration of the rays and directly as the cube of the index of refraction. Now we may consider this beam, of aggregate intensity IδSδω, to form an elementary filament of the radiation issuing in the direction of the normal from a perfect radiator. As such a body absorbs completely and therefore radiates equally in all directions in front of it, the total intensity of radiation from its element of surface δs is δs ∫I cos θδω), or δs.πI, while the volume-density of the total advancing, and receding radiation in front of it is 2V"fIdo.>, and therefore 4πV$^{−1}$I. If we take here Iδλ to represent the intensity between wave-lengths } and)+6}, this density is the quantity Eλ of which the temperature of the radiator is a function. Thus the quantity I-which optically is a measure of the brightness of the beam, and is conserved along it to the extent that, μ$^2$I is the same from whichever of its cross-sections the beam is supposed to be emitted—also determines its temperature, the latter being that of an enclosure containing undirected radiation of the same range δλ which is density E$_λ$δλ given by Eλ＝4πV$^{−1}$I, where V is the velocity of radiation in the enclosure. When a beam of radiation travels without suffering absorption, its temperature thus continues to be that of its source multiplied by the coefficient of emission of the source for that kind of radiation, this coefficient being less than unity except in the case of a perfect radiator; but when its intensity I falls by δI in any part of its path owing to absorption or other irreversible process, this involves a further fall of temperature of the energy of the beam and a rise of entropy which can be completely determined when the relation connecting μ$^{−3}$E$_λ$ with T and λ is known. Any directed quality in radiant energy increases its effective temperature. Splitting a beam into two at s reflecting and refracting surface diminishes the temperature of each part; it is true that if the reflecting surface were non molecular the operation could be reversed, but actually the reversed rays would encounter the reflecting molecules in different collocations, and could not (§ 11) recombine into the same detailed phase-relations as before. The direct solar radiation falling on the Earth is almost completely convertible into mechanical effect on account of its very high temperature; there seems ground for believing that certain constituents of it can actually be almost wholly turned to account by the ​ green leaves of plants. But the same -solar radiation, when broken up into diffused sky light, which has no definite direction, has fallen into equilibrium with a much lower temperature, through loss of its reversibility. It has been remarked that the temperatures of the planets can be roughly compared by means of this principle, if their coefficients of absorption of the solar radiation are assumed; that of Neptune comes out below −200° C., if we suppose that it is not kept higher by a supply of internal heat.
To obtain dynamical precision in this discussion an exact definition of the narrow beam such as is usually called a ray is essential. It can be specified as a, narrow filament of radiation, such as may be isolated within an infinitely thin, impermeable, bounding tube without thereby producing any disturbance of the motion. If either the tube or the surrounding radiation were not present to keep the beam in shape, it would spread sideways, as in optical diffraction. But the function of the tube is one of pure constraint; thus the change of energy content of a given length of the tube is represented by energy flowing into it at the end where the radiation enters, and leaving it at the other end, but with no leakage at the sides. The total radiation may be considered as made up of such filaments.
9. Temperature of the Sun.—The mean temperature of the radiating layers of the Sun may be estimated from Stefan's law, by computing the intensity of the radiation at his surface from that terrestrially observed, on the basis of the law of inverse squares; the result is about 6500° C. The application of Wien's law, which makes the wave-length of maximum energy vary inversely as the temperature, for the case of a perfectly radiating source, gives a. result 5 500° C. These numbers will naturally differ because (i) the Sun is not a perfect radiator, the constitution of his radiation in fact not following the law of that of a black body, (ii) the various radiating layers have different temperatures, (iii) the radiation may be in part due to chemical and electrical causes, and in so far would not be determined by the temperature alone. The fair agreement of these two estimates indicates, however, that the radiation is largely regulated by the temperature, that the layers from which the main part of it comes are at temperatures not very different, and that not very much of the complete radiation established in these layers and emitted from them is absorbed by the overlying layers.
10. Fluorescence.—When radiation of certain wave-lengths falls on a fluorescent body, it is largely absorbed, but in such manner as directly to excite other radiation of different type which is emitted in addition to the true temperature-radiation of the body. The distinction involved is that the latter radiation is spontaneously convertible with the heat of the absorbing body at its own temperature, without any external stimulus or compensation; it is, in fact, on the basis of this convertibility that the thermodynamic relations of the temperature-radiation have been established. According to the experimental law of Stokes, the wave-lengths of the fluorescent radiation are longer than those of the radiation which excites it. If the latter were directly transformed, in undiminished amount, into the fluorescent kind, this is what would be expected. For such a spontaneous change must involve loss of availability; and, beyond the wave-length of maximum energy in the spectrum, the temperature of a given density of radiation is greater the shorter its wave-length, as it is a function of that density and the wave-length alone such that greater radiation always corresponds to higher temperature. *But it would appear that the opposite should be the case for radiation of long wavelengths, lying on the other side of the maximum, in which the tendency would thus be for spontaneous change into shorter waves; this may perhaps be related to the fact that the lines of longer wave-lengths in spectra often come out brighter at lower temperatures, for they are then thrown on the other side of the maximum and cannot be thus degraded. The principle does not, however, have free play in the present case, even when the incident radiation is diffused and so has not the abnormally high temperature associated with a directed beam (§ 8), since part of it might be degraded into low-temperature heat, or there might be other compensation of chemical type for any abnormally high availability that might exist in the fluorescent radiation. It has been found that fluorescent radiation, showing a continuous or banded spectrum, can be excited in many gases and vapours; milky phosphorescence of considerable duration, and thus doubtless associated with chemical change, is produced in vacuum tubes, containing oxygen or other complexly constituted gases, by the electric discharge.
11. Entropy of a Ray.—If each definitely constituted beam of radiation has its own temperature and everything is reversible as above, a question arises as to the location of the process of averaging which enters into the idea of temperature. The answer can depend only on the fact, that although the beam is definite as to wave-length and intensity, yet it is far from being a simple wave-train, in that it is constituted of trains of limited lengths and various phases and polarizations, coming from the independent radiating molecules. When such a beam has once emerged, it travels without change, and can be reflected back intact to its source, and is in so far reversible; but when it has arrived there, the molecules of the source will have changed their positions, and it cannot be Wholly reabsorbed in the same manner as it was emitted. There must thus be some feature in the ultimate averaged constitution of the beam, emitted from a body in the definite steady state of internal motion determined by its temperature, which adapts it for spontaneous uncompensated re absorption into a body at its own (or a lower) temperature, but not at a higher one.
The question of the determination of the form of the function φ in § 6 would thus' appear to be closely connected with the other problems, hitherto imperfectly fathomed, relating to the statistics of kinetic molecular theory. A very interesting attack on the problem from this point of view has recently been made in various forms by Planck. It of course suffices to examine some simple type of radiating system, and the results will be of general validity. He considers an enclosure filled with radiation involving an entirely arbitrary succession of phases and polarizations along each ray, and also containing a system of fixed linear electric oscillators of the Hertzian type, which are taken' to represent the transforming action of radiating and absorbing matter. The radiation contained in the enclosure will be passed through these oscillators over and over again, now absorbed, now radiated, and each constituent will thus settle down in a unilateral or irreversible manner towards some definite intensity and composition. But it does not appear that a system of vibrators of this kind, each with its own period, can perform one of the main functions of a material absorber, namely, the transformation of the relative intensities of the various types of radiation in the enclosure to those corresponding to a common temperature. There would be equilibrium established only between the mean internal vibratory energy in the vibrators of each period and the density of radiation of that period; there is needed also some means of interchanging energy between vibrators of different periods, which probably involves doing away with their fixity, or else employing more complex vibrators and assuming a law of distribution of their internal energy. In the absence of any method of introducing this temperature equilibrium directly, Planck originally sought, in the case of each, independent constituent, for a function of its intensity of energy and its wave-length, restricted as to form by a certain assumed molecular relation, which has the property of continually increasing after the manner of entropy, during the progress of that constituent of the radiation in such a system towards its steady state. If the actual entropy S per unit volume could be thus determined, the relation of Clausius δS=δE/T would supply the connexion between the temperature and the density of radiant energy E. This procedure led him, in an indirect and tentative manner, to a relation d$^2$S/dE$^2$=−α/E, so that S=−αE logβ E, where α, B are functions of M an expression which conducts through Clausius's relation to E= (eβ)$^{−1}$e$^{−1/αT}$. ​The previous argument then gives E(λ,T)δλ=c$_1$λ$^{−5}$e$^{−c/λT}$δλ, a type of formula which was originally suggested by Wien on the basis of the analogy that it assigns the same distribution for the radiant energy, among the various frequencies of vibration, as for the energy of the molecules in a gas among their various velocities of translation. But the experimental inadequacy of this formula afterwards suggested a new procedure, as infra.
Processes may be theoretically assigned for the direct continuous transformation of radiant into mechanical energy. Thus we can imagine a radiating body at the centre of a wheel, carrying oblique vanes along its circumference, which reflect the radiation on to a ring of parallel fixed vanes, which finally reverse its path and return it to the centre. The pressure of the radiation will dr1ve the wheel, and in case its motion is not resisted, a very great velocity may be theoretically obtained. The thermodynamic compensation in such cases lies in the reduction of the effective temperature of the portion of the radiation not thus used up. We might even do away with the radiating body at the centre of the wheel, and consider a beam of definite radiation reflected backwards and forwards across a diameter. It is easy to see that its path will remain diametral; the work done by it in driving the wheel will be concomitant with increase of the wave-length, and therefore with expansion of the length occupied by the beam. The thermodynamic features are thus analogous to those of the more familiar case of an envelope filled with gas, which can change its thermal energy into mechanical energy by expansion of the envelope against mechanical resistances. In the case of the expanding gas pv = 2/3E$_0$, where E$_0$ is the total translator energy of the molecules, while in adiabatic expansion p=kv−γ. Thus the work gained in unlimited expansion, ∫pdv, is 2/3E$_0$/(γ−1). The final temperature being absolute zero, this should by Carnot’s principle be equal to the total initial energy of the gas that is in Connexion with temperature, constitutive energy of the molecules being excluded; when ~y-1 is less than § there is thus internal thermal energy in the molecules in addition to the translator energy. In the case of the beam of radiation, of length l, between n and n+δn reflexions, where δn is an integer, its total energy E is by § 2 reduced according to the law δE/E=−4cvδn/(c+v)$^2$. Also ¥=%L; thus %E-= -5% When 11 is small compared with c, this gives E=»<l“2; and p is then 2E/l, so that fpdl=E, the temperature of the beam being ultimately reduced to absolute, zero by the unlimited expansion. This is in accord with Carnot’s principle, in that the whole energy of the beam travelling in a vacuum is mechanically available when reduction to absolute zero of temperature is in our power.
12. Experimental Knowledge.—Under the stimulus of Wien’s investigation and of improvements in the construction of linear thermophiles and bolometers for the refined measurement of the distribution of energy along a spectrum, the general character of the curve connecting energy and wave-length in the complete radiation at a given temperature has been experimentally ascertained over a wide range. At each temperature there is a wave-length hm of maximum radiation, which is displaced towards the ultra-violet as the temperature rises, and Wien's law of homology (§ 6) shows that λ$_m$T should be constant. This deduction, and the law of homology itself, as also the law of Stefan and Boltzmann that the total radiation varies as T$^4$, have been closely verified by the experiments of Rubens and Kurlbaum, Lummer and Pringsheim, Paschen and others. They established a steady held of radiation inside a material enclosure by raising the Walls to a definite temperature, and measured the radiant intensity emitted from it through an opening or slit in the walls, by means of a bolometer or thermophile, this being the radiation of the so-called perfectly black body. The principle here involved formed one of the foundations of Balfour Stewart's early treatment of the theory, and had already been employed by him and Stokes (1860) in experiments on the polarized emission from tourmaline: cf. Stokes, Math. and Phys. Papers, iv. 136. It has been remarked by Planck and by Thiesen that the coefficient of T4 in Stefan's law, and the value of λ$_m$T, are two absolute physical constants independent of any particular kind of matter, which in conjunction with the constant of gravitation would determine an entirely absolute system of physical units. The form of the function φ(Tλ) adopted by Wien and in Planck’s earlier discussions, namely, c$_1$e$^{−c/Tλ}$, was found to agree fairly with experiment over the range from 100° C., to 1300° C., when c$_1$=1·24×10$^{−5}$, and c= 1·4435 in c.g.s. measure, but not so well when the range is farther extended: it appeared that a larger value of c was needed to represent the radiation for high values of Th, that is, for high temperature or for very long wave-lengths. Thiesen proposed the somewhat more general form c, (T})'°e“'/TA, and suggested that the value k=1/2 agrees better with the experimental numbers than Wien's value k=0; Lord Rayleigh was led (Phil. Mag., June 1000) towards this form with k equal to' unity from entirely different theoretical considerations, on the assumption of the Maxwell-Boltzmann distribution of the energy of a system, consisting of an isolated block of aether, among its free periods of vibration, infinite in number; in some cases this form appeared to give as good results as Wien's own.
Acting on a suggestion advanced by Lord Rayleigh, Rubens and Kurlbaum soon afterwards widely extended the test of the formulae by means of the so-called Reststrahlen. A substance such as an aniline dye, which exhibits selective absorption of any group of rays, also powerfully reflects those rays; and Rubens has been able thus to isolate in considerable purity the rays belonging to absorption bands very far down in the invisible ultra-red, having wave-length of order 10$^{−3}$ cm., which are intensely absorbed by substances such as sylvine, by means of five or six successive reflexions of the beam of radiation. By experiments ranging between temperatures −200° C. and + 1500° C. of the source of radiation, it has been found that the intensity of this definite radiation tends to vary simply as T, with close approximation, thus increasing indefinitely with the temperature, whereas Wien’s formula would make it tend to a definite limit. The only existing formula (except the one suggested by Lord Rayleigh) that proved to be in accord with this result was a new one advanced shortly before and supported on theoretical grounds by Planck, namely, Ei6l= Cλ$^{−5}$/(e'/"T- 1), which for small values of XT agrees with Wien's original form, known to be there satisfactory, While for larger values it tends towards C /c .}“4T; the new formula is, in fact, the simplest and most likely form that satisfies these two conditions. The point of Lord Rayleigh's argument was that, at any rate at low frequencies, the law of distribution would suggest an equable partition of the energy between temperature heat and radiant vibrations, and that therefore the energy of the latter should ultimately vary as T; and this prediction, which has thus been verified, may be grafted on to any formula that is in other respects appropriate.
Recognizing that his previous hypothesis, restricting the nature of the entropy in addition to its property of continually increasing, had thus to be abandoned, Planck had in fact made a fresh start on the basis of a train of ideas which was introduced by Boltzmann in 1877, in order to obtain a precise 'physical conception of entropy. According to the latter, for an indefinitely numerous system of molecules, with known properties and in given circumstances, there is a definite probability of the occurrence of each statistical distribution of velocities, or' say each “complexion” of the system, that is formally possible when all velocities consistent with given total energy are considered to be equally likely as regards each molecule; the distribution of greatest possible probability is the state of thermal equilibrium of the system, and the probability of any other state is a function of the entropy of that state. This conception can be developed only in very simple cases; the application to an ideal monatomic gas-system led Boltzmann to take the entropy proportional to the logarithm of the probability. This logarithmic law is in fact demanded in advance by the principle that the entropy of a system should be the sum of the entropies of its parts. By means of a priori considerations of this nature, referring to the distribution of internal vibratory energy among a system of linear electric vibrators of given period, and its equilibrium of exchanges with the surrounding radiant energy, Planck has been guided to an expression for the law of dependence of the entropy of that system on the temperature, which corresponds to the form of the law of radiation above stated. The result gains support from the fact that the expressions for the coefficients to which he is led give determinations of the ​ 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 λ$_m$ of maximum radiation, whether obeying Wien's law λ$_m$T=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 ​ indicates, moreover, that the true absorption bands in a gas of sufficiently low density must be extremely narrow. There is direct evidence that many of the more permanent gases do not sensibly emit light on being subjected to high temperature alone, when chemical action is excluded, while others give in these circumstances feeble continuous spectra; in fact, looking at the matter from the other side, the more permanent gases are very transparent to most kinds of radiation, and therefore must be very bad radiators as regards those kinds. The dark radiation of tiames has been identified with that belonging to the specific radiation of their gaseous products of combustion. There is thus ground for the view that the impacts of the colliding molecules in a gas, or rather their mutual actions as they swing sharply round each other in their orbits during an encounter, may not be sufficiently violent to excite sensibly the free vibrations of the definite periods belonging to the molecules. But they may produce radiation in other ways. While the velocity of an electron or other electric charge is being altered, it necessarily sends out a stream of radiation. Now the orbital motions of the electrons in an actual molecule must be so adjusted, as appears to be theoretically possible, that it does not emit radiation when in a steady state and moving with constant velocity. But in the violent changes of velocity that occur during an encounter this equipoise will be disturbed, and a stream of radiation, without definite periods, but such as might constitute its share of the equilibrium thermal radiation of the substance, may be expected while the encounter lasts. At very high temperatures the energy of this thermal radiation in an enclosure entirely overpowers the kinetic energy of the molecules present, for the former varies as T$^4$, while the latter measures T itself when the number of molecules remains the same. The radiation which can be excited in gases, confined as it is to extremely narrow bands in the spectrum, may indeed be expected to possess such intensity as to be thermally in equilibrium with extremely high temperatures. That the same gases absorb such radiations when comparatively cold and dark does not, of course, affect the case, because emissive and absorptive powers are proportional only for incident radiations of the intensity and type corresponding to the temperature of the body. Thus if our adiabatic enclosure of § 3 is prolonged into a tube of unlimited length which is filled with the gas, then when the temperature has become uniform that gas must send back out of the tube as much radiation as has passed down the tube and been absorbed by it; but if the tube is maintained at a lower temperature, it may return much less. The fact that it is now possible by great optical dispersion to make the line-spectra of prominences in the middle of the Sun’s disk stand out bright against the background of the continuous solar
spectrum, shows that the intensities of the radiations of these prominences correspond to a much higher temperature than that of the general radiating layer underneath them; their luminosity would thus seem to be due to some cause (electric or chemical) other than mere temperature. On the other hand, the general reversing gaseous layer which originates the dark Fraunhofer lines is at a lower temperature than the radiating layer; it is only when the light from the lower layers is eclipsed that its own direct bright-line spectrum flashes out. It is not necessary to attribute this selective flash-spectrum to temperature radiation; it can very well be ascribed to fluorescence stimulated by the intense illumination from beneath. When the radiation in a spectrum is constituted of wide bands it may on these principles be expected to be in equilibrium with a lower temperature than when it is constituted of narrow lines, if the total intensity is the same, in the cases compared; this is in keeping with the easier excitation of band spectra (cf. the banded absorption spectra), and with the fact that various gases and vapours do appear to emit band spectra more or less related to the temperature.
15. Constitution of Spectra.—In the problem of the unravelling of the constitutions of the very complex systems of spectral lines belonging to the various kinds of matter, considerable progress has been made in recent years. The beginning of definite knowledge was the discovery of Balmer in 1885, that the frequencies of vibration (n) of the hydrogen lines could be represented, very closely and within the limits of error of observation, by the formula n ∝ 1 − 4m$^2$, when for m is substituted the series of natural numbers 3, 4, 5, . . 15. Soon afterwards series of related lines were picked out from the spectra of other elements by Liveing and Dewar. Rydberg conducted a systematic investigation on the basis of a modification of Balmer’s law for hydrogen, namely, n=n$_o$−N/(m+μ)$^2$. He found that in the group of alkaline metals three series of lines exist, the so-called principal and two subordinate series. whose frequencies fit approximately into this formula, and that similar statements apply to other natural groups of elements; that the constant N is sensibly the same for all series and all substances, while n$_o$, and μ have different values for each; and that other approximate numerical relations exist. In each series the lines of high frequency crowd together towards a definite limit on the more refrangible side; near this limit they would, if visible, constitute a band. The principal or strongest series of lines shows reversal very readily. The lines of the first subordinate series are usually nebular, while those of the second subordinate or weakest series are sharp; but with a tendency to broaden towards the less refrangible side.
In most series there are, however, not more than six lines visible: helium and hydrogen are exceptions, no fewer than thirty lines of the principal series of the latter having been identified, the higher ones in stellar spectra only. But very remarkable progress has recently been made by R. W. Wood, by exciting fluorescent spectra in a metallic vapour, and also by applying a magnetic field to restore the lines sensitive to the Zeeman effect after the spectrum has been cut off by crossed nicols. The large aggregates of lines thus definitely revealed are also resolved by him into systems in other ways; when the stimulating light is confined to one period, say a single bright line of another substance, the spectrum excited consists of a limited number of lines equidistant in frequency, the interval common to all being presumably the frequency of some intrinsic orbital motion of the molecule. In this way the series belonging to some of the alkali metals have been obtained nearly complete.
Simultaneously with Rydberg, the problem of series was attacked by Kayser and Runge, who, in reducing their extensive standard observations, used the formula n=A+Bm$^{−2}$+Cm$^{−4}$, higher terms in this descending series being presumed to be negligible. This cannot be reconciled with Rydberg’s form, which gives on expansion terms involving m$^{−3}$; but for the higher values of m the discrepancies rapidly diminish, and do not prevent the picking out of the lines, the frequency-differences between successive lines then varying roughly as the inverse squares of the series of natural numbers. For low values of m neither mode of expression is applicable, as was to be expected; and it remains a problem for the future to ascertain if possible the rational formula to which they are approximations. More complex formulas have been suggested by Ritz and others, partly on theoretical grounds.
Considered dynamically, the question is that of the determination of the formula for the disturbed motions of the system which constitutes the molecule. Although we are still far from any definite line of attack, there are various indications that the quest is a practicable one. The lines of each series, sorted out by aid of the formulae above given, have properties in common: they are usually multiple lines, either all doublets in the case of monad elements, or generally triplets in the case of those of higher chemical valency; in very few cases are the series constituted of single lines. It is found also that the components of all the double or triple lines of a subordinate series are equidistant as regards frequency. In the case of a related group of elements, for example the alkaline metals, it appears that corresponding series are displaced continually towards, the less re frangible end as the atomic weight rises; it is found also that the interval in frequency between the double ​lines of a series diminishes with the atomic weight, and is proportional to its square. These relations suggest that the atomic weight might here act in part after the manner of a load attached to a fundamental vibrating system, which might conceivably be formed on the same plan for all the metals of the group; such a load would depress all the periods, and at the same time it would split them up in the manner above described, if it introduced dissymmetry into the vibrator. The discovery of Zeeman that a magnetic field triples each spectral line, and produces definite polarizations of the three components, in many cases further subdividing each component into lines placed usually all at equal intervals of frequency, is explained, and was in part predicted, by Lorentz on the basis of the electron theory, which finds the origin of radiation in a system of unitary electric charges describing orbits or executing vibrations in the molecule. Although these facts form substantial sign-posts, it has not yet been found possible to assign any likely structure to a vibrating system which would lead to a frequency formula for its free periods of the types given above. Indeed, the view is open that the group of lines constituting a series form a harmonic analysis of a single fundamental vibration not itself harmonic. If that be so, the intensities and other properties of the lines of a series ought all to vary together; it has in fact been found by Preston, and more fully verified by Runge and others, that the lines are multiplied into the same number of constituents in a magnetic field, with intervals in frequency that are the same for all of them. When the series consists of double or triple lines the separate components of the same compound line are not affected similarly, which shows that they are differently constituted. The view has also found support that the different behaviours of the various groups of lines in a spectrum show that they belong to independent vibrators. The form of the vibration sent out from a molecule into the aether depends on the form of the aggregate hodograph of the electronic orbits, which is in keeping with Rayleigh’s remark that the series-laws suggest the kinematic relations of revolving bodies rather than the vibrations of steady dynamical systems.
According to Rydberg, there is ground for the view that a natural group of chemical elements have all the same type of series spectrum, and that the various constants associated with this spectrum change rapidly in the same directions in passing from the elements of one group to the corresponding ones of the following groups, after the manner illustrated in graphical representations of Mendeléeff’s law by means of a continuous wavy curve in which each group of elements lies along this same ascending or descending branch; the chemical elements thus being built up in a series of types or groups, so that the individuals in successive groups correspond one to one in a regular progression, which may be put in evidence by connecting them by transverse curves. Illustrations have been worked out mathematically by J. J. Thomson of the effect of adding successive outer rings of electrons to stable vibrating collocations.
The frequencies of the series of very close lines which constitute a single band in a banded spectrum are connected by a law of quite different type, namely, in the simpler cases n$^2$= A−Bm$^2$. It may be remarked that this is the kind of relation that would apply to a row of independent similar vibrators in which the neighbours exert slight mutual influence of elastic type. If ξ denote displacement and x distance along the row, the equation d $^2$ξ/dt$^2$+k$^2$ξ=−gd $^2$ξ/dx$^2$ would represent the general features of their vibration, the right-hand side arising from the mutual elastic influences. If the ends of the line of vibrators, of length l, are fixed, or if the vibrators form a ring, the appropriate type of solution is ξ ∞ sin μx sin pt, where; μl = mπ and m is integral; further − p$^2$+k$^2$=gμ$^2$, hence p$^2$=k$^2$−gπ$^2$/l $^2$m$^2$, which is of the type above stated. Dynamical systems of this kind are illustrated by the Lagrangean linear system of connected bodies, such as, for example, a row of masses fixed along a tense cord, and each subject to a restoring elastic force of its own in addition to the tension of the cord. A single spectral line might thus be transformed into a band of this type, as the effect of disturbance arising from slight elastic connexions established in the molecule between a system of similar vibrators. But the series in line-spectra are of entirely different constitution; thus for the series expressed by the formula p$^2$=p$_0$$^2$−Bm$^{−2}$ the corresponding period-equation might be expressed in some such form as sin k(p$^2$−p$_0$$^2$)$^{−1/2}$=constant, which belongs to no type of vibrator hitherto analysed.
Authorities.—The experimental memoirs on the constitution of radiation are mostly in the Annalen der Physik; references are given by P. Drude, Lehrbuch der Optik, Leipzig, 1900; cf. also reports in the collection issued by the International Congress of Physics, Paris, 1900. See also Lord Rayleigh’s Scientific Papers, in various connexions; and Larmor, in Brit. Assoc. Reports, 1900–1902, also the Bakerian Lecture, Roy. Soc. Proc., 1909, for a general discussion of molecular statistical theory in this connexion. Planck’s Theorie der Wärmestrahlung, 1906, gives a discussion from his point of view; there is a summary by Wien in Ency. Math. Wiss. v. (3) pp. 282-357; also a lecture of H. A. Lorentz to the Math. Congress at Rome, 1908, and papers by J. H. Jeans, Phil. Mag., 1909, on the partition of energy. In spectrum analysis Kayser’s extensive treatise is the standard authority. Winckelmann’s Handbuch der Physik, vol. ii. (by Kayser, Drude, &c.), may also be consulted. (J. L.*)