August 13, 1891] 



NATURE 



357 



But in oblique collision between two not previously vibrating 

 doublets, even if the masses of shell and globule are equal, we 

 have a somewhat troublesome problem to tind the interval be- 

 tween the two impacts, when there arc tiuo, and to find the final 

 resulting vibration. When the component relative motion 

 parallel to the tangent plane of the first impact exceeds a certain 

 value depending on the radius of the outer surface of the shell, 

 the period of free vibration of the doublets, and the relative 

 velocity of approach ; there is no second impact, and the 

 doublets separate with no relative velocity perpendicular to 

 the tangent plane, but each with the energy of that component 

 of its previous motion converted into vibrational energy. When 

 the mass of the shell is much smaller than the mass of the 

 interior globule, almost every collision will consist of a large 

 number of impacts. It seems exceedingly difficult to find how to 

 calculate true statistics of these chattering collisions, and arrive 

 at sound conclusions as to the ultimate distribution of energy in 

 any of the very simplest cases other than Maxwell's original 

 case of i860 ; but, if the Boltzmann-Maxwell generalized doc- 

 trine is true, we ought to be able to see its truth as essential, 

 with special clearness in the simplest cases, even without going 

 through the full problem presented by the details. I can find 

 nothing in Maxwell's latest article on the subject (Camb. Phil. 

 Trans., May 6, 1878), or in any of his previous papers, proving 

 an affirmative answer to the question of § 7. 



(9) Going back to § 6, let the globules be initially distributed 

 as nearly as may be homogeneously through the hollow ; let 

 each globule be connected with neighbours by massless springs ; 

 ard let all the globules which are near the inner surlace of the 

 shell be connected with it also by massless springs. Or let 

 any number of smaller shells be inclosed within our outer 

 shell, and connected by massless springs, as represented by 

 the accompanying diagram, taken from a reprint of my Bal- 

 timore Lectures now in progress. Let two such outer shells, 



given at rest with their systems of globules in equilibrium within 

 them, be connected by massless springs, and be started in 

 motion, as were the shells of § 6. There will not now be the 

 great loss of energy from the vibration of the shells which there 

 was in § 6. On the contrary, the ultimate average kinetic 

 energy of the whole two hundred million million globules will be 

 certainly small in comparison with the ultimate average kinetic 

 energy of the single shell. It may be because each globule of 

 § 6 is free to wander that the energy is lost from the shell in 

 that case, and distributed among them. There is nothing vague 

 in their motion allowing them to take more and more energy, 

 now when they are connected by the massless springs. If we 

 suppose the motions infinitesimal, or if, whatever their ranges 

 may be, all forces are in simple proportion to displacements, the 

 elementary dynamical theorem oi fundamental viodes shows how 

 to find determinately each of the 600 million million and six 

 simple harmonic vibrations, of which the motion resulting from 

 the prescribed initial circumstances is constituted. It tells us 

 that the sum of the potential and kinetic energies of each mode 

 remains always of constant value, and that the time-average of 

 the changing kinetic energy during its period is half of this 

 constant value. Without fully solvin^j the problem for the 600 

 million million and six co-ordinates, it is easy to see that the 

 gravest fundamental mode of the motion actually produced in 

 the prescribed circumstances differs but little in period and 

 energy from the single simple harmonic vibration which the two 

 shells would take if the globules were rigidly connected to them, 

 or were removed from within them, and the other initial 

 circumstances were those of § 6. But this conclusion de- 

 pends on the forces being rigorously in simple proportion to 

 displacements. 



(io)i In no real case could they be so, and if there is any 

 deviation from the simple proportionality of force to displace- 



' Sections 10 10 17 added July 10, 1891. 

 NO. II 3 7, VOL. 44] 



ment, the independent superposition of motions does not hold 

 good. We have still a theorem of fundamental modes, although, 

 so far as I know, this theory has not yet been investigated. For 

 any stable system moving with a given sum, E, of potential and 

 kinetic energies, there must in general be at least as many 

 fiindaviental modes of rigorously periodic motion as there are 

 freedoms (or independent variables). But the configuration of 

 each fundamental mode is now noK generally similar for different 

 values of E ; and superposition of different fundamental modes, 

 whether with the same or with different values of E, has now 

 no meaning. It seems to me probable that every fundamental 

 mode is essentially unstable. It is so if Maxwell's fundamental 

 assumption^ " that the system, if left to itself in its actual state 

 of motion, will, sooner or later, pass through every phase which 

 is consistent with the equation of energy " is true. It seems to 

 me quite probable that this assumption is true, provided the 

 "actual state of motion " is not exactly, as to position and 

 velocity, a configuration of some one of the fundamental modes 

 of rigorously periodic motion, and provided also that the 

 " system " has not any exceptional character, such as those in- 

 dicated by Maxwell for cases in which he warns" us that his 

 assumption does not hold good. 



(11) But, conceding Maxwell's fundamental assumption, I do 

 not see in the mathematical workings of his paper ' any proof 

 of his conclusion " that the average kinetic energy correspond- 

 ing to any one of the variables is the same for every one of the 

 variables of the system." Indeed, as a general proposition its 

 meaning is not explained, and seems to me inexplicable. The 

 reduction of the kinetic energy to a sum of squares ■* leaves the 

 several parts of the whole with no correspondence to any de- 

 fined or definable set of independent variables. What, for 

 example, can the meaning of the conclusion ^ be for the case of 

 a jointed pendulum ? (a system of two rigid bodies, one sup- 

 ported on a fixed horizontal axis and the other on a parallel axis 

 fixed relatively to the first body, and both acted on only by 

 gravity). The conclusion is quite intelligible, however (but is 

 it true?), when the kinetic energy is expressible as a sum of 

 squares of rates of change of single co-ordinates each multiplied 

 by a function of all, or of some, of the coordinates.* Con- 

 sider, for example, the still easier case of these coefficients 

 constant. 



(12) Consider more particularly the easiest case of all, motion 

 of a single particle in a plane ; that is, the case of just two in- 

 dependent variables, say x, y ; and kinetic energy equal to 

 \{X- -V jf"-). The equations of motion are 



df'' 



dV (fiy ^__ dV 



dx dt'^ (ly ' 



where V is the potential energy, which may be any function of 

 ! X, y, subject only to the condition (required for stability) that it 

 j is essentially positive (its least value being, for brevity, taken as 

 zero). It is easily proved that, with any given value, E, for the 

 sum of kinetic and potential energies, there are two determinate 

 modes of periodic motion ; that is to say, there are two finite 

 closed curves such that, if m be projected from any point of 

 either with velocity equal to v^[2(E - V)] in the direction, either- 

 wards, of the tangent to the curve, its path will be exactly that 

 curve. In a very special class of cases there are only two such 

 periodic motions, but it is obvious that there are more than two 

 in other cases. 



(13) Take, for example, 



V = ^(a"-x^ + j8->- + cx^y^). 

 For all values of E we have 

 X = a cos (oU-e) 

 y = O 



as two fundamental modes. When E is infinitely small we have 

 only these two ; but for any finite value of E we have clearly 

 an infinite number of fundamental modes, and every mode diflfers 

 infinitely little from being a fundamental mode. To see this, 

 let m be projected from any point N in OX, in a direction per- 

 pendicular to OX, with a velocity equal to v^(2E-a-0N-). 



' "Scientific Papers," vol. ii. p. 714. ^ fii'd., pp. 714, 715. 



3 /fitW., pp. 716-726. 4 Hid., p. 722. 



5 Or ol Maxwell's " I," in p. 723. 



6 [It may be untrue for one set of co-ordinates, though true for others. 

 Consider, for example, uniform motion in a circle. For all systems of recti- 

 lineal rectangular co-ordinates (j-, >), time-av. a-2 = time-av. y-; but for 

 polar co-ordinates (r, V) we have not time-av. r- equal to time-av. r^-.^ 



! W. T., July 21, 1891.] 



^'^d^ = °cos()3/-/)} 



