ON ODK KNOWLEDGE OF THERMODYNAMICS. 91 



vided that the absolute temperature of a body is proportional to the mean 

 kinetic energy of its molecules taken over a quasi-period of their motion. 

 But to complete the proof it would still be necessary to show that a 

 quantity proportional to the mean kinetic energy of the molecules fulfils 

 the properties of temperature stated in § 3. The investigations on this 

 point will be considered in Section III. 



The hypothesis that the quasi-period i is very short compared with, 

 the time required to communicate a finite quantity of energy through the 

 molecules is tacitly involved in our regarding 8Q as a small variation. 

 On this hypothesis the value of T will vary very slowly, and T may 

 therefore be regarded as a continuously varying function. Hence, in 

 considering what takes place over a considerable number of quasi-periods, 

 we may replace the sign of summation by that of integration, and thus 

 obtain 



J;t"^t~ °^,T, 



the suffixes 1, 2 referring to the initial and final state of the body. 



14. The hypotheses involved in the definition of the quasi-period i by 

 means of equation (13) call for some comment. In his paper ' On a New 

 Mechanical Theorem relating to Stationary Motions,' ' CJausius gives a 

 rather more general form of the theorem, in which he supposes that there 

 may be different quantities i corresponding to different molecular co- 

 ordinates ; but in this case it seems to be necessary, according to him, 

 that in the varied motion all the i's shall be altered in the same ratio. If 

 such is assumed to be the case, S log i will be the same for all. Hence 

 we shall obtain for the portion whose quasi-period is i 



8Q=2T8 1ogi-|-28T, 



and, therefore, for the whole body 



28Q=22T . S log i-|-282T ; 



or, if we remove the signs of summation and let the quantities refer to 

 the entire system, 



8Q=2T81ogi + 28T, 

 whence 



^^=28(logiT) (14) 



as before. 



If we assume that each molecular coordinate {x, for example) always 

 fluctuates in the same periodic time i, so that the corresponding velocity 

 X vanishes at the times i,, ^j+i, <i+2i, &c., then the relation defining 

 the corresponding t, 



r -]'='>+' 

 mxhx =0, 



will be satisfied identically, and there will be no difficulty about the 

 matter. When, however, the molecular motions do not possess even this 

 amount of periodicity, Clausius gets over the difficulty by arguments of 

 the following general nature : — If we are dealing with a body of finite 



' Phil. Mag. vol. xlvi. (1873), p. 236. 



