356 



NATURE 



[August 13, 1891 



magnitudes in an interesting and important examination of the 

 subject in §§ 19, 20, 21 of his paper "On the Foundations of 

 the Kinetic Theory of Gases "(Trans. R.S.E. for May 1866). 



(2) Boltzmann, in his "Studien iiber das Gleichgewicht der 

 lebendigen Kraft zwischen bewemen materiellen Punkten " 

 {Sitzb. K. Akad. Wien, October 8, 1868), enunciated a large 

 extension of this theorem, and Maxwell a >till wider generaliza- 

 tion in his paper " On Boltzmann's Theorem on the Average 

 Distribution of Energy in a System of Material Points " (Cam- 

 bridge Phil. Soc. Trans., May 6, 1878, republished in vol. ii. of 

 Maxwell's "Scientific Papers," pp. 713-41), to the following 

 effect (p. 716) :— 



" In the ultimate state of the system, the average kinetic 

 energy of two given portions of the system must be in the ratio 

 of the number nf degrees of freedom of those portions." 



Much disbelief and doubt has been felt as to the complete 

 tru^h, or the extent of cases for which there is truth, of this 

 proposition. 



(3) For a test case, differing as little as possible from Max- 

 well's original case of solid elastic spheres, consider a hollow 

 spherical shell and a solid sphere — globule we shall call it for 

 brevity— within the shell. I must first digress to remark that 

 what has hitherto by Maxwell and Clausius and others before 

 and after them been called for brevity an "elastic sphere," is 

 not an elastic solid, capable of rotation and of elastic deforma- 

 tion ; and therefore capable of an infinite number of modes of 

 steady vibration, into which, of finer and finer degrees of nodal 

 subdivision and shorter and shorter periods, all translational 

 energy would, if the Boltzmann-Maxwell generalized proposition 

 were true, be ultimately transformed by collisions. The, 

 "smooth elastic spheres" are really Boscovich point-atoms, 

 with their translational inertia, and with, for law of force, zero 

 force at every distance between two points exceeding the sum of 

 the radii of the two balls, and infinite repuMon at exactly this 

 distance. We may use Boscovich similarly for the hollow shell 

 with globule in its interior, and so do away with all question as 

 to vibrations due to elasticity of material, whether of the shell or 

 of the globule. Let us simply suppose the mutual action 

 between the shell and the globule to be nothing except at an 

 instant of collision, and then to be such that their relative com- 

 ponent velocity along the radius through the point of contact is 

 reversed by the collision, while the motion of their centre of 

 inertia remains unchanged. 



(4) For brevity, we shall call the shell and interior globule of 

 § 3, a double molecule, or sometimes, for more brevity, a 

 doublet. The "smooth elastic sphere" of § 3 will be called 

 simply an atom, or a single atom ; and the radius or diameter 

 or surface of the atom will mean the radius or diameter or 

 surface of the corresponding sphere. (This explanation is 

 necessary to avoid an ambiguity which might occur with re- 

 ference to the common expression "sphere of action" of a 

 Boscovich atom.) 



(5) Consider now a vast number of atoms and doublets, 

 inclosed in a perfectly rigid fixed surface, having the property 

 of reversing the normal component velocity of approach of any 

 atom or shell or doublet at the instant of contact of surfaces, 

 while leaving unchanged the absolute velocity of the centre of 

 inertia of the two. Let any velocity or velocities in any direc- 

 tion or directions be given to any one or more of the atoms or 

 of the shells or globules constituting the doublets. According 

 to the Boltzmann-Maxwell doctrine, the motion will become 

 distributed through the system, so that ultimately the time- 

 average kinetic energy of each atom, each shell, and each 

 globule shall be equal ; and therefore that of each doublet 

 double that of each atom. This is certainly a very marvellous 

 conclusion; but I see no reason to doubt it on that account. 

 After all, it is not obviously more marvellous than the seemingly 

 well-proved conclusion that in a mixed assemblage of colliding 

 single atoms, some of which have a million million times the 

 mass of others, the smaller masses will ultimately average a 

 million times the velocity of the larger. But it is not included in 

 Maxwell's proof for single atoms of different masses [(34) of his 

 " I>ynamical Theory of Gases" referred to above] ; and the 

 condition that the globules inclosed in the shells are prevented 

 by the shells from collisions with one another violates Tail's 

 condition [(C) of § 18 of " Foundations of K. T. Gases "], " that 

 there is perfectly free access for collision between each pair of 

 particles whether of the same or of different systems." An 

 independent investigation of such a simple and definite case as 

 that of the atoms and doublets defined in §§ 3-5 is desirable as a 



test, or would be interesting as an illustration were test not 

 needed, for the exceedingly wide generalization set forth in the 

 Boltzmann-Maxwell doctrine. 



(6) Next, instead of only a single globule within the shell of 

 § 4, let there be a vast number. To fix ideas let the mass of the 

 shell be equal to a hundred times the sum of the masses of the 

 globules, and let the number of the globules be a hundred 

 million million. Let two such shells be connected by a push- 

 and-pull massless spring. Let all be given at rest, with the 

 spring stretched to any extent ; and then left free. According 

 to the Boltzmann-Maxwell doctrine, the motion produced 

 initially by the spring will become distributed through the 

 system, so that ultimately the sum of the kinetic energies of the 

 globules within each shell will he a hundred million million 

 times the average kinetic energy of the shell. The average 

 velocity^ of the shell will ultimately be a hundred-millionth of 

 the average velocity of the globules. A corresponding proposi- 

 tion in the kinetic theory of gases is that, if two rigid shells, each 

 weighing i gram, and containing a centigram of monatomic gas, 

 be attached to the two prongs of a massless perfectly elastic 

 tuning-fork, and set to vibrate, the gas will become heated in 

 virtue of its viscous resistance to the vibration excited in it by 

 the vibration of the shell, until nearly all the initial energy of 

 the tuning-fork is thus spent. 



(7) Going back to the double molecules of § 5, suppose the 

 internal globule to be so connected by massless springs with the 

 shell that the globule is urged towards the centre of the shell 

 with a force simply proportional to the distance between the 

 centres of the two. This arrangement, which I gave in my 

 Baltimore Lectures, in 1884, as an illustration for vibratory 

 molecules embedded in ether, would be equivalent to two masses 

 connected by a massless spring, if we had only motions in one 

 line to consider ; but it has the advantage of being perfectly iso- 

 tropic, and giving for all motions parallel to any fixed line 

 exactly the same result as if there were no motion perpendicular 

 to it. When a pair of masses connected by a spring strikes a 

 fixed obstacle or a movable body, with the line of their centres 

 not exactly perpendicular to the tangent plane of contact, it is 

 caused to rotate. No such complication affects our isotropic 

 doublet. An assemblage of such doublets being given moving 

 about within a rigid inclosing surface, will the ultimate sta- 

 tistics be, for each doublet,^ equal average kinetic energies of 

 motion of centre of inertia, and of relative motion of the two 

 constituents? 



(8) If we try to answer this question synthetically, we find a 

 complex and troublesome problem in the details of all but the 

 very simplest case of collision which can occur, which is direct 

 collision between two not previously vibrating doublets, or any 

 collision of one not previously vibrating doublet against a fixed 

 plane. In this case, if the masses of globule and shell are 

 equal, a complete collision consists of two impacts at an interval 

 of time equal to half the period of free vibration of the doublet, 

 and after the second impact there is separation without vibration, 

 just as if we had had single spheres instead of the doublets. 



' The " average velocity of a particle," irrespectively of direction, is (in 

 the kinetic theory of gases) a convenient expression for the square root of 

 the time-average of the square of its velocity. 



^ This i.nplies equal average kinetic energies of the two constituents ; and, 

 conversely, equal average kinetic energies of the two constiruents, except in 

 the case of their masses being equal, implies the equality stated in the text. 

 Let u, ji be absolute component velocities of two masses, in, ;«', per- 

 pendicular to a fixed plane; U the corresponding component velocity of 

 their centre of inertia ; and r that of their mutual relative motion. We 

 have 



:U - 



:U + 



(t) 



whence 



)>iH^ - m H - =(m - m ) U^ - -„ — ^ \jr. . . (2) 



Now suppose the time-average of Ur to be zero. In every ca e in which 

 this is so, we have, by (2), 



Time-av. i^/tifi - m'u-^^-{in - m) X Tiraeav. [ U2 - """ ^" ^ | . (3) 



Hence in any case in which 



Time-av. ;««- = Time-av. m'ti"' (4) 



I we have 



?') X Time-av. \ U^ 



(S) 



and therefore, except when : 



Time-av. (;« -j- w')U2 = Time-av- 



(6) 



which proves the propisition, because, as we readily see from _(i), 

 \vtm'r^l(m -{- in) is, in every case, the kinetic ene-gy of the relative, 

 motions, n — U, and U - n. 



NO. 1137, VOL. 44] 



