96 Mr. E. P. Culverwell on the 



become transformed [by repeated collisions] into vibrational 

 energy of higher and higher nodal subdivisions, if each mole- 

 cule is a continuous elastic solid." If this were true there is, I 

 think, no doubt that the proof would involve also a proof that 

 Boltzmamr's permanent configuration is that which all dyna- 

 mical systems will finally assume. 



Barely touching on the difficulty, dwelt on by Sir W. 

 Thomson at Montreal, that elastic molecules would not on his 

 theory, or indeed on Boltzmann's theory either, possess the 

 known properties of a gas, I will take as an example the 

 only case in which the equations of motion can be integrated 

 in finite terms, i. e. that in which the stress between two par- 

 ticles varies directly as the distance. Taking attractive action, 

 each particle moves harmonically round the centre of Mass 

 of the system, and the mutual actions of the particles do not 

 on the whole tend to distribute the energy equally among all 

 the degrees of freedom. For instance, let there be a system 

 comprising any number, great or small, of balls or points 

 following this law, the balls being in motion with, say, very 

 small velocities. Let two of the particles be given very 

 violent blows in opposite directions, so that the total momentum 

 of the system is unaltered, then the interaction of the balls 

 will not cause any of the energy given to the two balls to be 

 shared among the other balls. Observe that the particles must 

 either be mathematical points or else able to move through 

 each other, but this makes no difference to my argument, 

 which is that there is nothing in the nature of dynamical 

 equations in virtue of which their solutions tend to a perma- 

 nent average distribution of energy. . 



This instance shows that it is impossible to prove in general 

 that a set of particles will tend to the Boltzmann configura- 

 tion, in which the energy is equally distributed among all 

 the degrees of freedom (though it does not show that they 

 would not do so for certain laws of force other than the 

 direct distance) ; and it shows also that the assumption 

 commonly made, that permanent states are independent of 

 initial conditions, is really not based on dynamical grounds 

 but on physical experience, and that it involves some 

 principle analogous to the second law of thermodynamics. 



But surely no instance is required to show the impossibility 

 of proving that, whatever be the initial conditions, the effect of 

 repeated intermolecular collisions is to equalize the energy 

 among the different degrees of freedom. For the well-known 

 property of reversal, which all complete and purely dynamical 

 systems fulfil, shows that for every configuration which tends 

 to an equal distribution of energy, there is another which 



