70 Dr. W. F. Gr. Swann on Equipartition of 



way, the continuity of flow of the representative points from 

 the normal to the abnormal regions of the generalized space 

 may be preserved without the time during which the points 

 remain in the abnormal regions being constrained to be very 

 short, and without there being any congestion of the points 

 such as would be contrary to the Hamiltonian laws as repre- 

 sented by Liouville's theorem of differential invariance. The 

 ideas in this connexion may perhaps be made clearer by a 

 genera] consideration of the kind of motion which a point 

 must have in the generalized space if it is to correspond to 

 such a system as a piece of copper. 



To fix our ideas, let us for a moment suppose that the 

 positions and momenta of the positive and negative elec- 

 trons are proper generalized coordinates in terms of which 

 to specify the system. In our piece of copper we have a 

 condition of affairs where, viewed from a somewhat crude 

 standpoint, the electrons are all describing comparatively 

 fixed orbits ; or perhaps it would be better to say that the 

 average constancy in general feature peculiar to the material 

 as a whole is retained not so much by the orbits remaining 

 fixed as by a condition of affairs in which a change of one 

 kind in one direction in one atom (change of orbital radius,. 

 or expulsion of an electron, for example) is on the average 

 balanced by a change of the reverse kind in some other atom. 

 Individual coordinates may be changing continually, and this 

 provides a means by which the point in the generalized space 

 may travel along continuously, and not be confined to the 

 neighbourhood of one spot. If at any instant we fix on 

 the coordinates associated with one atom, find the two- 

 dimensional coordinate planes, in the generalized space, 

 containing the axes of these coordinates, and project the 

 motion of the point in the generalized space on these planes, 

 we shall get a certain group of curves. If we do the same 

 thing for the other atoms we shall find that the group of 

 curves is repeated in its general characteristics. This pos- 

 sibility of picking out, from the whole set of coordinate 

 planes of the complete system, groups of coordinate planes 

 such that the projection of the representative point on one 

 group is similar to the projection on other groups, sym- 

 bolizes the similarity of the atoms. 



The permanence of the system is not necessarily symbolized, 

 however, by each of these groups of projections maintaining 

 its form permanently (forming re-entrant paths) , for an elec- 

 tron, for example, may leave one atom and become part of 

 another, so that what were formally the coordinates of one 

 atom may in a little while be distributed in a very complex 



