212 SCIENCE PROGRESS. 



escape from the difficulty : In the proof of the law jT /!( x + T) 

 we assumed that the final co-ordinates and momenta q\ . . . 

 P'„ were functions of the initial co-ordinates and momenta 

 and of no other quantities whatever. But suppose the 

 molecule contains a class of co-ordinates n, . . . a,., with 

 corresponding momenta (5 T . . . /3 r , which are not functions 

 at all of the other co-ordinates and momenta </,... P„, but 

 entirely independent of them. If that be so ^ + T in the 

 expression r h(x + T) is the energy of the qp co-ordinates and 

 momenta only, and will not include the a/3 system. It is true 

 that the afi set may be amenable to the same treatment, 

 and we might perhaps prove that for them also the law of 

 distribution is denoted by £ -* ( x' + T '> where ■% ls the potential, 

 T' the kinetic energy of the system afi. But we cannot 



piuvc LiidL n ■■= k. nnu su luc wliu — ^ need not 



depend on the number of co-ordinates of the q set and the 

 a set respectively. On this assumption we may make 



111 \X 2 A- 1' 2 A- Z\ ^ 



„^r + -, if n and T relate to all the co-ordinates 



2 1 11 



both of the q and a sets. 



The hypothesis cannot be unreasonable, because all 

 writers on the kinetic theory of gases have made it once, 

 when they ignore the rotation of their elastic spheres. The 

 three angular velocities of a smooth sphere behave to the 

 translation velocities exactly in the same way as /3 to/ on 

 the above hypothesis. And a system in which the trans- 

 lation velocities are distributed according to the law t~ hu \ 

 and the angular velocities according to e~ kK '\ would, if the 

 spheres be smooth, be in equilibrium with h + k. But the 

 following objection appears to be fatal : If the temperature 

 of the gas depends on the qp system alone, including the velo- 

 cities of translation, you must on this hypothesis admit that 

 of two equal quantities of gas at the same pressure and 

 temperature, one might contain more energy than the other 

 because it might contain more of the a/3 sort. If temperature 

 depends on both forms of energy, then one must, under some 

 circumstances, be capable of conversion into the other, 



