286 Mr PocMington, On the Kinetic Theory of Matter. 



from the results of physical experiments, but in such experiments 

 the phases reached by a system may not include all those that are 

 geometrically possible. For example, they certainly will not if 

 each atom includes a gyrostat. There are two ways in which 

 it is possible for the systems ultimately to differ, namely their 

 physical properties may, on passing from one system to another, 

 change either continuously, or in an essentially discontinuous 

 manner. 



If the change is continuous, we should expect to find evidence 

 of its existence in nature. If, for example, a continuous change 

 from the argon atom to the mercury atom is geometrically possible, 

 each form being stable, we should expect to find the intermediate 

 forms occurring (and those near to either limit should occur in 

 nearly as great abundance as the limiting forms), even if we could 

 not pass artificially from form to form. As they do not occur, we 

 conclude that they cannot. The fact that the atomic weights 

 of the known elements form an orderly series of numbers also 

 suggests strongly that the various kinds of atom form an essentially 

 discrete group. 



On the other hand, it is not possible for the systems to differ 

 ultimately in a discontinuous manner. For let each system be 

 represented by a point in a 2?t-dimensional space where q 1 ... 

 p x ... are the coordinates of the point. Initially these points fill 

 a 2w-dimensional space (depending on the limits within which the 

 energies of the systems lie) with uniform density. At any sub- 

 sequent time the points lie within the same space, and, from the 

 Jacobian equation, their density of distribution is the same as 

 before. Hence they still fill the space, and the phases form a 

 continuous group. The physical properties of the systems can 

 only differ on account of differences in phase, and hence must vary 

 continuously if at all. 



6. We see then that the acceptance of Assumption B, even if 

 we admit that the alteration of an atom is geometrically possible, 

 involves consequences that directly contradict the observed phe- 

 nomena, and we must now reject this assumption and investigate 

 the case where there are one or more linear relations between the 

 velocities. If these are integrable, they lead to relations between 

 the coordinates in virtue of which the number of independent 

 coordinates is reduced, and we fall back on the case already con- 

 sidered. We hence suppose the equations not to be integrable. 

 Such equations occur in the simpler cases of rolling motion. One 

 consequence of the existence of such equations is that the number 

 of independent coordinates exceeds that of the independent velo- 

 cities. In particular cases, it still happens that the Jacobian 

 of the independent velocities (or of linear functions of them) and 

 the coordinates with respect to their original values is unity. This 



