ON OUIl KNOWLEDGE OF THERMODYNAMICS. 105 



states of the molecules of a body individually from those which define the 

 state of the molecules in the aggregate. 



It was stated in § 24 that, under certain circumstances, a polycyclic 

 system may possess exactly the same properties as a monocyclic system, 

 even though the coordinates defining the circulating motions of the 

 system are all independent. The system considered by J. J. Thomson 

 belongs to this class, for the necessary conditions are secured by the 

 assumption which the author makes in the following statement concerning 

 the kinetic energy due to the molecular or ' unconstrainable ' coordinates 

 «i of the system : ' — If the term 



^[(uti,)u-+ . . . } 



involves any ' controllable ' coordinate <f), then it is evident that this co- 

 ordinate <f> must enter as a factor into all the terms in the form expressed 

 by the equation 



i{(«w)"'^+ . . . ]=yW{(uicyn^-+ . . . } . . (60) 



where the coefficients (iiuy do not involve ^, otherwise the phenomenon 

 would be influenced more by the motion of some particular molecule than 

 by that of others.'^ In other words, the investigation is limited in its 

 application to the thermal properties of a single body, for in the case of a 

 system of more than one body it is. evident that the phenomena would 

 be difi'erently influenced by the motion of the molecules in diff'erent 

 bodies. In such a case the molecular kinetic energy of each individual 

 body would contain a common factor f(<j>), which might be different for 

 difi'erent bodies. Even in the case of a single body the assumption, 

 thoiTgh plausible, can hardly be regarded as axiomatic. 



The other assumptions involved in J. J. Thomson's work are similar 

 to those of Helmholtz, but they impose fewer restrictions on the gene- 

 rality of the proof. While Helmholtz assumes that the changes in the 

 state of the system take place so slowly that the velocities of the con- 

 trollable coordinates (q„ or ^) do not enter into the energy of the system, 

 Thomson merely assumes that the portions of the kinetic energy due to 

 the controllable and molecular coordinates are distinct, so that the whole 

 kinetic energy is of the form 



T=T, + T„ (61) 



where the part T„ alone is to be taken as the dynamical analogue of 

 temperature, the part T^^^ denoting the kinetic energy due to motions of 

 the body as a whole and other controllable motions. 



Moreover, Thomson only assumes that the potential energy of the 

 system is a function of the controllable and not of the molecular coordi- 

 nates, so that 



^^=^%^^ (62) 



and 



2^.> = (63) 



' Applications of Dynamics, pp. 94, 95. 



'' In comparing J. J. Thomson's proof with that of Helmholtz we must write 



