Energy and Radiation Theory. 75 



conclusion is that the total energy of the system must be 

 infinite " *. 



Now, quite apart from the question as to whether the 

 theorem of equipartition is justifiable as applied to an 

 infinite number of degrees of freedom, a point which it 

 is not proposed here to discuss, it seems that even in 

 assuming the theorem we ought to be very careful to 

 consider whether the quantity R which figures in the mea- 

 surements on gases is really the R, which ought to figure in 

 the expression RT/2 for the energy associated with one 

 degree of freedom above concerned. If it were true that 

 the energy were expressible as the sum of squared terms, 

 part of the sum involving the setherial modes of motion, 

 and part involving the velocity coordinates of the centres 

 of gravity of the molecules, the argument would indeed 

 be justifiable; but nobody would maintain that the inter- 

 action between the radiation and the gas was capable of 

 being discussed completely in terms of a sort of interaction 

 between the waves and the centres of gravity of the mole- 

 cules. If we picture minutely the phenomenon of two 

 gases coming into temperature equilibrium by means of 

 radiation alone, we are constrained to imagine electronic 

 orbits and so forth in the molecule, with the various parts 

 mutually influencing each other. The radiation from the 



* The argument is usually developed by considering the case of a box 

 with perfectly reflecting Avails inclosing pure aether only. It may be of 

 interest to point out that there is a fundamental difference between the 

 meanings to be attached to a coordinate associated with the setherial 

 vibrations and such a coordinate as one of the ordinary coordinates of a 

 gas molecule for example. The number of harmonic modes comprised 

 between certain wave-length limits in a box of volume V is half the 

 number comprised between the same wave-length limits in a box of 

 volume 2V ; but the increased number of coordinates in the larger box 

 is not obtained by a duplication of the coordinates in the smaller box, as 

 it would be in the cases of two boxes containing ordinary gas molecules, 

 and in which the coordinates concerned were the ordinary coordinates of 

 the gas molecules. It is obtained rather by an interpolation of extra 

 wave-lengths which were not capable of existence in the smaller box. 

 The scale of the Fourier analysis is. in fact, more fine-grained in the 

 larger box than in tiie smaller. These considerations are only of 

 subsidiary importance, however, for the mathematical analysis takes no 

 account of the nature of the coordinates. Again, although in the sense 

 above explained, the quality of the radiation in the box of volume 2V 

 may be said to be mathematically different from that in the box of 

 volume V, that which is observed in any optical instrument, viz. the 

 energy density of the radiation comprised between assigned ranges of 

 wave-length, is the same for each case, the average amplitude of the 

 harmonic mode* in the larger box being smaller than the average 

 amplitudes of the smaller number of harmonic modes in the smaller 

 box. 



