ON OUR KNOWLEDGE OF THERMODYNAMICS. 101 



their molecular coordinates only (not between their controllable co- 

 ordinates) in such a manner that the motions of the two systems are 

 individually unaffected by the coupling, but that the coupled system 

 forms a single monocyclic system. Corresponding to equality of 

 temperature we must have equality between two integrating divisors of 

 (7Q for the two monocyclic sj'stems, and tliese integrating divisors must 

 always remain equal so long as the two systems are coupled together. 



Such being the conditions imposed upon the problem from thermal 

 considerations, Helmholtz investigates the general form of the integrating 

 •divisors for two monocyclic systems in order that this condition may be 

 fulfilled — i.e., that equality of these divisors may bo the criterion of 

 the possibility of coupling the systems. This kind of coupling he calls 

 ' isoniorous.' As simple instances of such coupled dynamical systems 

 the following are mentioned : — 



(i.) Two revolving wheels may be coupled together by joining their 

 axles if their angular velocities are equal. If either wheel carries a 

 Watt's governor or centrifugal regulator in which the distance of the 

 revolving balls from the axis is controllable, the angular velocities of the 

 two wheels can thus be equalised just as two bodies may be brought to 

 the same temperature by applying suitable pressures. 



(ii.) Two circulating streams of liquid in annular vessels can be com- 

 bined into a single stream wherever their linear velocities are identical, 

 and the necessary conditions may be secured by suitably varying the 

 form and dimensions of the containing vessels. 



The principle of limited availability when heat is converted into 

 work by reversible processes depends on the impossibility of controlling 

 the individual molecules of a body : all that we can do is to commu- 

 nicate heat to the body by placing it in contact with another body, which 

 must be at the same temperature if the process is to be reversible. Cor- 

 responding to this property we must make the hypothesis that in a 

 monocyclic system it is impossible to operate directly on the gyrostatic 

 coordinates by means of external forces, but that work can only be 

 communicated through these coordinates by coupling the system with 

 another monocyclic system, and that the coupling must be ' isomorous.' 

 If this assumption be made, the monocyclic system will evidently possess 

 properties corresponding to the principle of limited availability. 



27. Let T/i and 172 be the required integrating divisors for tlie two 

 systems, so that whenever rji^r] and ^2='7 tlie systems can be coupled 

 together. Let the corresponding entropies be a-, and cto ; then for such a 

 coupled system we must have 



(ZQ,=7yr7o-i "1 



dq,=y,da., } . . . (44) 



.-. dq =dq^+dq.2=7]:i(<7i+<T.:^ J 



therefore 77 is an integrating divisor of (?Q for the entire coupled system. 



Any other integrating divisor will be the product of t; with an arbi- 

 trary function of the corresponding entropy (§ 2). But the kinetic 

 energies Ti, T2, Ti+Tg are integrating divisors of cZQ,, r7Q,, and (?Q 

 (since the coupled system is supposed to be monocyclic). Therefore 



T,-=»7,</.(^,) ) 



T,=r,,ilf(a,) I . . . . (45) 



