

1891.] Doctrine regarding Distribution of tinergy. 85 



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 assump- 

 tion* " 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 pro- 

 vided also that the " system " has not any exceptional character, 

 such as those indicated by Maxwell for cases in which he warnst 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 paperj any proof of his 

 conclusion " that the average kinetic energy corresponding 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 defined or definable set of inde- 

 pendent 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 supported 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, how- 

 ever (but is it true ?), when the kinetic energy is expressible as a 

 sum of squares of rates of change of single co-ordinates each multi- 

 plied by a function of all, or of some, of the co-ordinates.^[ Consider, 

 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 independent 

 variables, say, #, t/; and kinetic energy equal to ^(# 2 -f f). The 

 equations of motion are 



<Px_ _dV_ tfy__ _<ZV 



df~ dx' d?~ dy ' 



* " Scientific Papers," Yol. II, p. 714. 



t Ibid., pp. 714, 715. 



Ibid., pp. 716726. 



Ibid., p. 722. 



|| Or of Maxwell's " b," in p. 723. 



^[ [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 rectilineal rectangular 

 co-ordinates (x, y), time-av. # 2 = time-av. y z ; but for polar co-ordinates (r, 6) we 

 have not time-av. r 2 equal to time-av. r 2 b 2 . W. T., July 21, 1891.] 



