1 68 



NATURE 



[December 13, 1900 



quite independent of the bombardment. The reports from 

 numerous observers showed that as the showers moved along 

 over the earth's surface those in front of it reported that the 

 noise of the exploding dynamite occurred just before the shower ; 

 those in the wake of the shower reported that the shower came 

 before the explosion, while those in the midst of the shower, of 

 course, heard the explosion while it was raining. There was no 

 evidence that the explosion had any effect on the clouds. Careful 

 observations in Washington, D.C., during the whole of this 

 first experiment, and during subsequent experiments with 

 explosives, warranted the conclusion that no rainfall was 

 produced by bombardment. 



About that time a "rain wizard" commenced operating 

 in Ohio. His method consisted in locking himself in a 

 barn, house, luggage van or other room, wherein he made 

 a fire and burned or evaporated certain chemicals, the 

 smoke of which rose through the roof out of some impromptu 

 chimney or stove-pipe and dissipated itself in the thin air. 

 Of course jt was claimed that the chemicals exerted a great 

 influence on the atmosphere and forced rain to come. 

 Occasionally rain did come after one, two, or three days 

 of a chemical performance, but equally often it did not 

 come. The Weather Bureau was often importuned for advice 

 as to when the wizard should be called to any given town, 

 and whether the inhabitants would be justified in paying him 

 his fee of several hundred dollars. Eventually, a prominent 

 railway company rigged up a car for his use, and during the 

 years 1892-4 made it convenient for all the citizens on its 

 lines to invoke the aid of " the rain producer." Of course there 

 were numerous cases in which the operations were followed by 

 rain ; those who studied the Daily Weather Map could see at a 

 glance that these rains accorded with the general weather con- 

 ditions and had nothing to do with the rain-making operations. 

 So long as frequent rains occurred, although they were natural 

 and were predicted by the Weather Bureau on the basis of the 

 weather map from day to day, yet the farmers of Iowa, Kansas 

 and Nebraska, ignoring this fact, were sure to accredit all 

 success to the wizard. 



During the last great drought in California, 1898-1899, the 

 citizens of one city authorised an extensive and expensive 

 system of experiments by gases and by cannon, but were for- 

 tunately saved the necessity of actually wasting their money by 

 the fact that an abundant rain fell naturally just before they 

 were ready to begin their own operations. 



Occasionally we still receive newspaper items reviving the old 

 story that floods of rain were broken up by cannonading at 

 Rome, or that rain was produced by cannonading in Italy, or 

 that hailstorms were averted from a special vineyard that was 

 protected by lightning rods while neighbouring vineyards 

 suffered. These are all repetitions of the same old myths, or 

 repetitions of useless experiments, and the intelligent reader 

 may dismiss them as having no foundation. No matter how 

 severely his land may be suffering from drought or flood, he 

 should seek some other mode of relief and not waste his time 

 and money in efforts to change the nature of the clouds or the 

 a' mosphere. 



ON THE STATISTICAL DYNAMICS OF GAS 

 THEORY AS ILLUSTRATED BY METEOR 

 SWARMS AND OPTICAL RAYS> 



I MAGINE a cloud of meteors pursuing an orbit in space under 

 outside attraction — in fact, in any conservative field of force. 

 Let us consider a group of the meteors around a given central 

 one. As they keep together their velocities are nearly the same. 

 When the central meteor has passed into another part of the orbit, 

 the surrounding region containing these same meteors will have 

 altered in shape ; it will in fact usually have become much 

 elongated. If we merely count large and small meteors alike, 

 we can define the density of their distribution in space in the 

 neighbourhood of this group ; it will be inversely as the volume 

 occupied by them. Now consider their deviations from a mean 

 velocity, say that of the central meteor of the group ; we can 

 draw from an origin a vector representing the velocity of each 

 meteor, and the ends of these vectors will mark out a region in 

 the velocity diagram whose shape and volume will represent the 



1 A paper read by Dr. J. Larmor, F.R.S., before Section A of the British 

 Association at Bradford, September, 1900. 



NO. 1624, VOL. 63] 



character and range of deviation. It results from a very 

 general proposition in dynamics that as the central meteor 

 moves along its path the region occupied by the group of its 

 neighbours multiplied by the corresponding region in their 

 velocity diagram remains constant. Or we may say that the 

 density at the group considered, estimated by mere numbers, not 

 by size, varies during its motion proportionally 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 principle 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 ^wu^ + wW 

 conserved : hence if Suj be the range of velocity at any point in 

 the initial position, and 81^2 tbat at the corresponding point in any 

 subsequent position of the group, we have v-^v-^ = v.^v^, these 

 positions remaining unvaried and the variation being due to 

 different meteors passing through them. But if Sa>^ and Sqj.j are 

 the initial and final conical angles of divergence of the velocity 

 vectors, corresponding regions in the velocity diagram are of 

 extents Su^. v^'bw-^ and Suj- >'2^5(i'2 : 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 uSco, because v5u 

 remains constant. Now the number of meteors that cross per 

 unit time per unit area of a plane at right angles to the paih of 

 the central meteor is equal to this density multiplied by v : thus 

 here it remains proportional to v^Stw, as the central meteor moves 

 on. In the corpuscular formulation of geometrical optics this 

 result carries jthe general law that the concentration in cross- 

 section of a beam of light at different points of its path is pro- 

 portional 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 

 restriction 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 er-"^^, 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 fact, the chance of an encounter for 

 particles of energies E and E' will involve the product 

 g-hE^^hE' Qr ^— i>(E-t-E'), and an encounter does not alter this total 

 energy E + E' ; while the domains or extents of range of two col 

 h liding groups each nearly homogeneous and estimated, as above, 

 ' by deviation 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 product, 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 «— ''*^, 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 case each 

 value of the resultant velocity would occur with a frequency 



