100 REPORT — 1891. 



same changes will take place in a less time ; hence, if we use accented 

 letters for the original motion, we shall have generally 



q—aq 

 s-=.ns' 



(35) 



The effect of communicating a quantity of energy dQ, through the 

 speed coordinates of such a system will be to increase the rate of working 

 of the system, and therefore to increase n. 



Now we have 



= u'^{ruds\) + ndn^{ri,^^) . . . (36) 



But when the rate is constant, c7h=0 ; cZQ=0 ; 



.-. ;^(2V7s',)=0 .... (37) 



which defines the monocycle. 



/. dq = ndn^{q',/,) .... (38) 

 But 



.-. ^^Q=^^=2J(log«) . . . (39) 



T n 



The quantity corresponding to entropy — viz., 2 'log h — log (constant) } 

 differs from that given by the method of Ciausius, but the two investiga- 

 tions are easily reconciled. For writing (36) in the form 



dCl^ndn^{q\s\)+n''-\d^{(^,s\)-'^{s\dq:,)]=0 . (40) 

 the assumption made in Clausius' method is that 



5(-s'„%'0-0 (41) 



and under such circumstances 



dq,=ndn-2T +ri?d2T .... (42) 



cZQ^rfQ^2dri 2rZT' 

 •• T n'T T T'" 



=2'i(lognT')=2(ilog(T/») . . (43) 



which agrees with (14). 



26. By far the most interesting part of Helmholtz's papers is 

 ihat in which he has investigated the dynamical analogue of thermal 

 equilibrium between two or more bodies of equal temperature. Of this 

 portion we will now give a brief sketch. 



If two bodies of equal temperature are placed in contact, the state 

 of either body will be unafiected, and the system, taken as a whole, will 

 be subject to the two laws of thermodynamics. 



The dj'-namical analogue to be investigated is that of two monocyclic 

 systems coupled together by means of geometrical connections between 



