December J27, 1900] 



NA TURE 



217 



produce the motion of the body against pressure excited by the 

 surrounding radiation. The hypothesis of friction is now out 

 of court in ultimate molecular physics ; while the thermo- 

 dynamic bearing of a pressure produced by radiation has been 

 developed by Bartoli and Boltzmann (1884), and that of the 

 Doppler effect by Wien (1893). 



Application of the Doppler Principle. — The procedure of Wien 

 amounts to isolating a region of radiation within a perfectly 

 reflecting enclosure, and estimating the average shortening of 

 the constituent wave-lengths produced by a very slow shrinkage 

 of its volume. The argument is, however, much simplified if 

 the enclosure is taken to be spherical and to remain so ; for it 

 may then be easily shown that each individual undulation is 

 shortened in the same ratio as is the radius of the enclosure, so 

 that the undulatory content remains similar to itself, with 

 uniformly shortened wave-lengths, whether it is uniformly dis- 

 tributed as regards direction or not, and whatever its constitution 

 may be. But if there is a very small piece of a mateyial 

 radiator in the enclosure, the radiation initially inside will have 

 been reduced by its radiating and absorbing action to that 

 corresponding to its temperature. In that case the shrinkage 

 must retain it always, at each stage of its transformation, in the 

 constitution corresponding to some temperature. Otherwise 

 differences of temperature would be effectively established 

 between the various constituents of the radiation in the en- 

 closure ; these could be permanent in the absence of material 

 bodies ; but if the latter are present this would involve degrada- 

 tion of their energy, for which there is here no room, because, 

 on the principles of Stewart and Kirchhoff, the state correspond- 

 ing to given energy and volume and temperature is determinate. 

 Thus we infer that if the wave-lengths of the steady radiation 

 corresponding to any one temperature are all altered in the 

 same ratio, we obtain a distribution which corresponds to some 

 other temperature in every respect except absolute intensities. 



Direct Transformation of Mechanical Energy into Radiation J'- 

 — There is one point, however, that rewards examination. 

 When undulations of any kind are reflected from an advancing 

 wall, there is slightly more energy in the reflected beam than 

 there was in the incident beam, although its length is shorter 

 on account of the Doppler effect. This requires that the undu- 

 lations must oppose a resistance to the advancing wall, and that 

 the mechanical work required to push on the wall is directly 

 transformed into undulatory energy. In fact, let us consider 

 the mechanism of the reflexion. Suppose the displacement in a 

 directly incident wave-train, with velocity of propagation c, to 

 be | = a cos {mx-mct) ; that in the reflected train will be |' = 

 a' cos (m'x + m'ct), where a', m' are determined by the condition 

 that the total displacement is annulled at the advancing reflector, 

 because no disturbance penetrates beyond it ; therefore, when 

 x = vl, where v is its velocity, |-i-|' = o. Thus we must have 



a' — -a, and m' ■ 



in agreement with the usual state- 



ment of the Doppler effect when v is small compared with c. 

 Observe, in fact, that the direct and reflected wave-trains have 

 a system of nodes which travel with velocity v, and that the 

 moving reflector coincides with one of them. Now the veloci- 

 ties dijdt and d^'jdt in these two trains are not equal. Their 

 mean squares, on which the kinetic energy per unit length 

 depends, are as w- to m'^. The potential energies per unit 

 length depend on the means of (d^/dx)'' and {d^'/dxf, and are 

 of course in the same ratio. Thus the energies, .per unit length 

 in the direct and reflected trains are as ;«- to m''^, while the 

 lengths of the trains are as m' to m ; hence their total energies 

 are as m to m' ; in other words, the reflected train has received 

 an accretion of energy equal to i - m'lm of the incident energy, 

 which can only have come from mechanical work spent in 

 pushing on the reflector with its velocity v. The opposing 

 pressure is thus in numerical magnitude the fraction 



(fn'\ c 

 I ~ ~^)^ of the density of the incident energy, which works 



1 1 he present form of this argument arose out of some remarks con- 

 tributed by Prof. FitzGerald, and by Mr. Alfred Walker of Bradford, to 

 the discussion on this paper. Mr. Walker points out that by reflecting the 

 radiation [from a hot body, situated at the centre of a wheel, by a ring of 

 obliqu* vanes around its circumference, and then reversing its path by direct 

 reflexion from a ring of fixed vanes outside the wheel, so as to return it into 

 the source, its pressure may be (theoretically) utilised to drive the wheel, 

 and in time to get up a high speed if there is no load : the thermodynamic 

 compensation m this very interesting arrangement lies in the lowering of 

 the temperature of the part of the incident radiation that is not thus utilised. 



NO. 1626, VOL. 63] 



C'^ + v'^ 



direct and reflected, that is in front of the reflector. 



When V is small compared with c, this agrees with Maxwell's 

 law for the pressure of radiation. This case is also theoretically 

 interesting, because in the application to sether-waves | is the 

 displacement of the ?ether elements whose velocity d^/dt repre- 

 sents the magnetic force ; so that here we have an actual case 

 in which this vector |, hitherto introduced only in the theoretical 

 dynamics of electron-theory, is essential to a bare statement of 

 the facts. Another remark here arises. It has been held that 

 a beam of light is an irreversible agent, because the radiant 

 pressure at the front of the beam has nothing to work against, 

 and its work is therefore degraded. But suppose it had a 

 reflector moving with its own velocity c to work against ; our 

 result shows that the pressure vanishes and no work is done. 

 Thus that objection to the thermodynamic treatment of a single 

 ray is not well founded. 



This generalisation of the theory of radiant pressure to all 

 kinds of undulatory motion is based on the conservation of the 

 energy. It remains to consider the mechanical origin of the 

 pressure. In the special case of an unlimited stretched cord 

 carrying transverse waves the advancing reflector may be a 

 lamina, through a small hole in which the cord passes without 

 friction : the cord is straight on one side of the lamina, and in- 

 clined on the other side on account of the vibration ; and it is 

 easily shown that the resultant of the tensions on the two sides 

 provides a force acting on the lamina which, when averaged, 

 agrees with the general formula. In the case of an extended 

 medium with advancing transverse waves, which are reflected 

 directly, the origin of the pressure is not so obvious, because 

 there is not an obvious mechanism for a reflector which would 

 sweep the waves in front of it as it advances. In the asthereal 

 case we can, however, on the basis of electron-theory, imagine a 

 constitution for a reflector which will turn back the radiation on 

 the same principle as a metallic mirror totally reflects Hertzian 

 waves, and thus obtain an idea of how the force acts. 



The case of direct incidence has here been treated for sim- 

 plicity ; that of oblique incidence easily follows ; the expression 

 for the pressure is reduced in the ratio of the square of the cosine 

 of the angle of incidence. If we average up, after Boltzmann, 

 for the natural radiation in an enclosure, which is incident 

 equally at all angles, we find that the pressure exerted is one- 

 third of the total density of radiant energy. 



Adiabatics of an enclosed Mass of Radiation, and resulting 

 General Laws. — Now consider an enclosure of volume V con-; 

 taining radiant energy travelling indifferently in all directions, 

 and of total density E ; and let its volume be shrunk by 5V. 

 This requires mechanical work ^E5V, which is changed into 

 radiant energy : thus 



EV -1- JE5V= (E - 8E)(V - 8V), 



where E - 5E is the new density at volume V - 5V, This gives 

 |ESV = V5E, or Eoo V~*- 



As already explained, if the original state has the constitution 

 as regards wave-lengths corresponding to a temperature T, the 

 new state must correspond to some other temperature T - 8T. 

 Thus we can gain work by absorbing the radiation into the 

 working substance of a thermal engine at the one temperature, 

 and extracting it at the other ; as the process is reversible, we 

 have by Carnot's principle 



iE5V/EV=-5T/T, 

 so that Too V~i- 



Thus Eoc T*, which is Stefan's law for the relation of the 

 aggregate natural radiation to the temperature, established 

 theoretically on these lines by Boltzmann. 



Moreover, the Doppler principle has shown us that in the 

 uniform shrinkage of a spherical enclosure the wave-lengths 

 diminish as the linear dimensions, and therefore as V», or in- 

 versely as T by the above result. Thus in the radiations at 

 different temperatures, if the scale of wave-length is reduced 

 inversely as the temperature the curves of constitution of the 

 radiation become homologous, i.e., their ordinates are all in the 

 same ratio. This is Wien's theoretical law. 



These relations show that the energy of the radiation corre- 

 sponding to the temperature T, which lies between wave-lengths 

 \ and A + 5A, is of the form \~°f{\T)h\. The investigation, 

 theoretical (Wien, Planck, Rayleigh, &c.) and experimental 



