Energy and Radiation 1 heory. 69 



coordinates o£ the copper and rearranging their magnitudes 

 so as to correspond to other kinds of matter. The mathe- 

 matics includes in its survey every case in which the values 

 of the coordinates can be conceived as arranged to corre- 

 spond to states other than those of the elements of matter. 

 Thus for example, if we could take as our coordinates in a 

 piece of matter the ordinary spacial and momental coordi- 

 nates of the positive and negative electrons, supposing such 

 to exist, the mathematics involved in establishing the infinite 

 probability of the state given by (2), (3), &c, would insist 

 on considering in addition to all those cases corresponding to 

 the various species of matter, all those arrangements in which 

 the positions and velocities of the electrons were distributed 

 in more or less random fashion after the manner of gas 

 molecules. 



When we view matters in the above light, it would not 

 seem surprising if the w^hole of those arrangements which 

 we should be prepared to recognize as copper were asso- 

 ciated with only an infinitesimal proportion of the generalized 

 space corresponding to the prescribed energy-range. In fact, 

 is not the infinite probability of the state represented by (2), 

 (3), &c. dependent on the fact that the mathematics takes 

 account of all those distributions of energy among the 

 generalized coordinates which we more particularly asso- 

 ciate with the random gas molecule type ? 



Now in connexion with the above arguments, it might 

 be objected that although at some particular view, all the 

 representative points corresponding to the various copper 

 pieces referred to above might exist in one of the 

 "abnormal" regions of the generalized space, yet, since 

 the abnormal regions form only a small fraction of the 

 total volume of the portion of the space concerned, it will 

 not be long before a point has passed out of the abnormal 

 region into a normal one. It must be pointed out in this 

 connexion, however, that though the abnormal regions 

 corresponding to our piece of copper, for example, may 

 form an infinitesimal fraction of the volume of the available 

 space, they need not be of such a kind that all of the 

 coordinates are constrained to suffer but limited ranges of 

 variation in order to remain wdthin the region. Neither 

 is it necessary to assume that the practical permanence of 

 the copper, as copper, is to be represented by the coordinates 

 all lingering near certain definite values ; the abnormal 

 regions may in fact be distributed linearly over great dis- 

 tances in the space like a thin channel permeating hither and 

 thither in a solid block in three-dimensional space. In this 



