362 HERMANN VON HELMHOLTZ 



internal motion, and since Helmholtz considers the hypothesis 

 of Clausius and Boltzmann (that this is the case for all other 

 bodies also) to be highly probable, he next inquires into the 

 conditions under which vis viva becomes the integrating 

 denominator for monocyclic systems with fixed associations of 

 the moving elements as is the case for simple monocyclic 

 systems. He finds that it is a condition that the entropy of the 

 restricted system should be a homogeneous function of the 

 first degree of the momentum of the unrestricted system, 

 whence it results that if the complete system of parameters is 

 kept constant, the total momentum and the velocities of the 

 restricted system must increase in proportion to the resulting 

 momentum and the resulting velocity of the internal motion. 

 It was shown that all cases known to us at present of the 

 mechanical coupling of any pair of cyclical motions fulfil the 

 conditions under which vis viva is an integrating denominator 

 in the compound monocyclic system. He further succeeded in 

 defining the special mode of these fixed associations between 

 the moving parts of the system more exactly. When two 

 monocyclic systems originally independent of each other are 

 transformed by a proper adjustment of external forces into 

 a state corresponding with this particular kind of fixed associa- 

 tion, it is possible to bring them into this fixed association 

 without disturbance of the motion present ; and they can then, 

 upon further alterations of energy, continue their motion while 

 maintaining this fixed combination, which is again analogous 

 to the motion of heat, in which two bodies of equal temperature 

 can be brought into conducting contact without alteration of 

 their internal motions, so that during new and sufficiently slow 

 changes they maintain a constant temperature. This state of 

 temporary fixed association is termed by Helmholtz the coup- 

 ling of the system. He points out, as especially interesting, 

 the case in which a mechanical association is set up between 

 two systems which have equal values for one of their integrating 

 denominators, in such a way that, so long as this association 

 persists, the equality of this denominator will be maintained, as 

 is the case in the contact of two bodies at the same temperature, 

 where the temperature is the integrating denominator. Helm- 

 holtz calls this kind of association an isomerous coupling. It 

 is found universally that if monocyclic systems only admit of 



