TRANSACTIONS OF SECTION A. 633 



considered, estimated by mere numbers, not by size, varies during its motion pro- 

 portionally to the extent of the region on the velocity diagram which corresponds 

 to it. 



This is true whether mutual attractions of the meteors are sensibly effective or 

 not ; in fact, the generalised form of this proposition, together with a set of 

 similar ones relating to the various partial groups of coordinates and velocity 

 components, forms an equivalent of the fundamental law of Action which is the 

 unique basis of dynamical theory. 



Now, suppose that the mutual attractions are insensible, and that W is the 

 potential of the conservative field : then for a single meteor of mass m and velocity 

 V we have the energy -J-wzu- + 7nW conserved : hence if du^ be the range of velocity 

 at any point in the initial position, and Su, that at the corresponding point in any 

 subsequent position of the group, we have Vjfiuj = v.jSvo, these positions remaining 

 unvaried and the variation being due to different meteors passing through them. 

 But if dco^ and Su.j are the initial and final conical angles of divergence of the 

 velocity vectors, corresponding regions in the velocity diagram are of extents 

 fiuj.Vi'-Sci)! and 8v.yV.-r8o}.-, : these quantities are, therefore, in all cases proportional 

 to the densities at the group in its two positions. In our present case of mutual 

 attractions insensible, the volume density is thus proportional to vSco, because vSv 

 remains constant. Now the number of meteors that cross per unit time per unit 

 area of a plane at right angles to the path of the central meteor is equal to this 

 density multiplied by v : thus here it remains proportional to v'^Soi, as the central 

 meteor moves on. In the corpuscular formulation of geometrical optics this 

 result carries the general law that the concentration in cross-section of a beam of 

 light at different points of its path is proportional to the solid angular divergence 

 of the rays multiplied by the square of the refractive index, which is also directly 

 necessitated by thermodynamic principles ; as a special case it limits the possible 

 brightness of images in the well-known way. 



In the moving stream of particles we have thus a quantity that is conserved in 

 each group — namely, the ratio of the density at a group to the extent of the region 

 or domain on the velocity diagram which corresponds to it ; but this ratio may 

 vary in any way from group to group along the stream, while there is no restric- 

 tion on the velocities of the various groups. If two streams cross or interpenetrate 

 each other, or interfere in other ways, all this will be upset owing to the collisions. 

 Can we assign a statistical law of distribution of velocities that will remain 

 permanent when streams, which can be thus arranged into nearly homogeneous 

 groups, are crossing each other in all directions, so that we pass to a model of a 

 gas ? Maxwell showed that if the number of particles each of which has a total 

 energy E is proportional to e~''^ , where h is some constant (which defines the 

 temperature), while the particles in each group range uniformly, except as regards 

 this factor, with respect to distribution in position and velocity jointly, as above, 

 then this will be the case. In ^act, the chance of an encounter for particles of 

 energies E and E' will involve the product g-iiEg-hE' qj. g-h{E+E')^ g^^^ ^^^ encounter 

 does not alter this total energy E + E' ; while the domains or extents of ran^e of 

 two colliding groups each nearly homogeneous and estimated, as above, by devia- 

 tion from a central particle in position and velocity jointly, will have the same 

 product after the encounter as before by virtue of the Action principle. It 

 follows that the statistical chances of encounter, which depend on this joint pro- 

 duct, will be the same in the actual motion as are those of reversed encounter in 

 the same motion statistically reversed. But if the motion of a swarm with 

 velocities fortuitously directed can be thus statistically reversed, recovering its 

 previous statistics, its molecular statistics must have become steady ; in fact, we 

 have in such a system just the same distribution of encountering groups in one 

 direction as in the reverse direction : thus we have here one steady state. The 

 same argument, indeed, shows that a distribution, such that the number per unit 

 volume of particles whose velocity deviations correspond to a given region in the 

 velocity diagram, is proportional to the extent of that region without this factor 

 g-hE^ will also be a steady one. This is the case of equable distribution in each 

 group as regards only the position and velocity diagrams conjointly ; but in this 



