Jitly 26, 1877J 



NA TURE 



H5 



in one part of a vessel, which we shall call B, than when 

 it is in the part A, these two parts being separated by a 

 stratum, C, within which the potential varies continuously. 

 The medium consisting of the spheres jVwill be dense in 

 A, it will become rarer in the stratum C, and there will 

 be hardly any of these spheres in B. 



Now let the potential energy of a sphere of the set A^' 

 be much greater when it is in A than when it is in B, and 

 let it vary continuously from the one value to the other in 

 the stratum C. Then the spheres of this set will be 

 thickly scattered in B, will thin out in the stratum C, and 

 will be very rare in A. 



The two sets of spheres are thus kept in great measure 

 separate in A and B, while free to exchange their kinetic 

 energy by collisions within the stratum C. 



Now by definition, the temperatures of two bodies are 

 equal if, when the two bodies are placed in contact, their 

 thermal state remains the same. We cannot apply this 

 definition to the two sets of spheres in Prop. 11., for they 

 were inextricably mixed up together, but we have now 

 got them almost completely separated from each other 

 into two distinct regions. T..cy are therefore practically 

 distinct bodies, and we can test their temperatures 

 separately. 



Hence the statement, that the temperatures of two gases 

 are equal when the kinetic energy of the centre of mass 

 of a molecule is the same in each, is true, not only of gases 

 mixed together, but of two pure gases in different parts 

 of the same vessel. 



If we assume that a powerful external force acts on 

 each molecule tending to a fixed centre belonging to that 

 molecule, each molecule will always remain very near its 

 own fi.xed centre of force, and the assemblage of molecules 

 will behave like a solid body. But forces of this kind are 

 included among those considered in Prop. IV., so that 

 the relation between temperature and the kinetic energy 

 of the centre of mass of a single molecule must be extended 

 even to solids. 



Returning once more to the general" expression for the 

 average number of molecules in a group, we may make it 

 yield us information with respect to the average number 

 of sets of molecules which, at a given instant, are in a 

 given configuration with respect to each other. 



For instance, if two molecules act on each other, and if 

 X is the potential energy due to this action corresponding 

 to a distance ;', then the number of pairs of molecules 

 whose distance is between r and r -\- d r will be propor- 

 tional to r-^ " ''<. In the case of attraction, ^ is negative, 

 so that there will be a greater number of pairs of mole- 

 cules within these limits of distance than there would 

 have been if they did not attract each other. In the case 

 of repulsion, x is positive, so that the repulsion diminishes 

 the number of pairs within the distance of repulsion. If 

 the potential eneigy of a pair of molecules rapidly increases 

 to an enormous value when the distance between their 

 centres becomes less than a given quantity, the number 

 of pairs which are within the given distance will be 

 practically zero, and the molecules will behave like 

 smooth ligid-elastic spheres. 



By making the "molecule" include three or more 

 molecules, and making x the potential energy of this 

 system, we may extend the theorem to the simultaneous 

 encounter of three or more molecules, so that these cases. 



which were formally excluded in the earlier propositions, 

 do not in any way interfere with the absolute generality 

 of the final result. 



The clear way in which Mr. Watson has demonstrated 

 these propositions leaves us no escape from the terrible 

 generality of his results. Some of these, no doubt, are 

 very satisfactory to us in our present state of opinion 

 about the constitution of bodies, but there are others 

 which are likely to startle us out of our complacency, and 

 perhaps ultimately to drive us out of all the hypotheses in 

 which we have hitherto found refuge into that state of 

 thoroughly conscious ignorance which is the prelude to 

 every real advance in knowledge. 



If we know from observation either the specific heat of 

 a gas at constant pressure, or the ratio of its specific heats 

 at constant pressure and at constant volume, we can 

 determine the ratio of the rate of increase of its total 

 energy to the rate of increase of the energy of agitation of 

 the centres of its molecules. Now if the molecule has m 

 degrees of freedom, its total kinetic energy is to the 

 energy of agitation of its centre of mass as ni to 3. It 

 is probable that the internal potential energy of the 

 molecule increases as the temperature rises, and this 

 would make the ratio of the whole energy to that of 

 agitation of centres greater than that of m to 3, so that 

 if we know this ratio by experiment, we can assert that m 

 cannot exceed a certain value. 



For chlorine, ammonia, and sulphuretted hydrogen, m 

 cannot exceed 6 ; for hydrogen, oxygen, nitrogen, air, 

 carbonic oxide, nitrous cxide, and hydrochloric acid, it 

 cannot exceed 5, and for mercury gas, according to the 

 experiments of Kundt and Warburg, it cannot exceed 3. 



Now Boltzmann has pointed out in a paper : " Uber 

 die Natur der Gasmoleciile " (Vienna Acad., December 

 14, 1876), that if the molecules were rigid-elastic bodies of 

 any form, m would be 6, that if they were smooth figures 

 of revolution, the velocity of rotation about the axis of 

 figure would not be afiected by the collisions, so that m 

 would be 5, and that if they were smooth spheres, the 

 three component velocities of rotation would each of 

 them be independent of coUisions, so that m would be 

 reduced to 3, and these values are in striking agreement 

 with the phenomena of the three groups of gases. 



But before we accept this somewhat promising hypo- 

 thesis, let us try to construct a rigid-elastic body. It will 

 not do to take a body formed of continuous matter 

 endowed with elastic properties, and to increase the 

 coefficients of elasticity without limit till the body 

 becomes practically rigid. For such a body, though 

 apparently rigid, is in reality capable of internal vibra- 

 tions, and these of an infinite variety of types, so that 

 the body has an infinite number of degrees of freedom. 



The same objection applies to all atoms constructed of 

 continuous, non-rigid matter, such as the vortex-atoms of 

 Thomson. Such atoms would soon convert all their 

 energy of agitation into internal energy, and the specific 

 heat of a substance composed of them would be infinite. 



A truly rigid-elastic body is one whose encounters with 

 similar bodies take place as if both were elastic, but 

 which is not capable of being set into a state of internal 

 vibration. We must take a perfectly rigid body and 

 endow it with the power of repelling all other bodies, 

 but only when they come within a very short distance 



