102 KEPOET 1891. 



■whence 



x(<T,+cT,)=<l.(<T,)+iI;(cro) . . . (46) 



giving, on differentiating first with regard to cr^ and then with regard 



to 0-2, 



x"=o. 



Therefore on integration 



<f,=a + cai I . . . (47) 



ij/=h + C(r2 J 



But if s,, S.2 be the generalised momenta corresponding to the gyro- 

 static coordinates of the two systems, we have 



dQ2=2T,d\ogs,=-r}da.2} • • • • K'*°-^ 



From (45), (47), and (48) 



2rZ log .,= >'-) 



'^ + ''*^'L .... (49) 

 2dlogs,= /pJ 



.*. by integration, ^(a-i)=a + co-, = (si/a)^'^[ /'t;n\ 



where a, /3 are constants. Substituting in (45) we find 



.... (51) 





V-2 



These, then, are the most general forms of 17,, tj., possessing the two 

 qualifications by which temperature is characterised — namely, (i.) Carnot's 

 principle and (ii.) the property of defining the state of a body in relation 

 to its thermal equilibrium with another body. 



28. There is still another condition to be satisfied in finding a kinetic 

 analogue of temperature — namely, the property that if two bodies, A and 

 B, are in thermal equilibrium, and if A and C are also in thermal equi- 

 librium, then B and C will be in thermal equilibrium. 



This imposes on our monocyclic systems the condition that whenever 

 a system (1) can be coupled with either of two systems (2) and (3), the 

 systems (2) and (3) can also be coupled together. The examples already 

 given of wheels revolving with equal angular velocity and of circulating 

 streams are instances of the fulfilment of this condition. 



In all such cases the ge(<metrical equations connecting the coordinates 

 of the coupled bodies must be of the form 



^i='A2=X3 • • • • • (52) 



where <^i only involves the coordinates of the first body, i/^o those of the 

 second, and xs those of the third. 



Applying § 23, we see that if F (si, Sj) denote the entropy of the 



