ON OUR KNOWLEDGE OF THERMODYNAMICS. 107 



dynamically possible when the Lagrantrian function contains odd powers 

 of tha generalised velocities, and that this is the case when it has been 

 modified so that some of the velocities have been ignored owing to the 

 corresponding generalised momenta being constant. But this simply 

 means that the ignored velocities are not to be reversed when the motion 

 of the system is reversed. It is easy to see that in a dynamical 

 system it is not in general possible to reverse some of the motions 

 without reversing them all. 



Poincare now considers, as a test case, that in which the system is 

 acted on by no external forces, and he considers, more particularly, what 

 happens when the entropy is approaching its maximum, his object being 

 to discover whether there is any dynamical way of proving the funda- 

 mental thermodynamic property that the entropy of a system is con- 

 tinually increasing. If such is the case, then, taking S as the entropy, 

 dS/dt must always be positive. Now, taking E as the energy and adopt- 

 ing the notation of Helmholtz, the Hamiltonian equations give 



dp_dB d.s_ _8E_ 



di~ ds' dt dp' 



whence 



dS ^fdSdE_dSdE\ ^ggv 



df-^\dp ds ds'dpj ' ' ' * ^ ' 



In the subsequent investigation Poincare assumes that when the entropy 

 is a maximum the system must he in stable equilibrium, so that in this 

 condition of the system we have not only 



-9S=o and f=0, 



dp OS 



but also 



1^45=0 and ^=^=0. 



dt dp dt ab 



Such a step appears to me to be quite unjustifiable, for it amounts to 

 nothing less than assuming that the system under investigation is at the 

 absolute zero of temperature, and the entropy in such a case will of 

 course be infinite. 



If we have any number of bodies enclosed in an adiathermanous 

 envelope it is known from physical, not dynamical, considerations that 

 the entropy of the system will tend to a maximufii as the temperatures 

 of the various bodies become equalised, and yet when all the bodies are 

 at the same temperature the molecules are still in a lively state of motion, 

 not at rest, as in Poincare's investigation. 



It is also to be noted that Poincare nowhere makes use of the fact 

 that S is the entropy of the system. 



Hence it is difficult to see how Poincare's result can have any direct 

 bearing on the principle of degradation of energy or even how it can have 

 a thermodynamical interpretation at all. 



34. At the same time, there are many considerations which render it 

 prima facie unlikely that the monocyclic method should be capable of 

 accounting for the principle of degradation of energy. 



A system which is irreversible will certainly not be monocyclic 

 according to the definition of Helmholtz, and hence we cannot assume 



