716 BOLTXMAHN'S THEOREM ox THE AVERAGE DISTRIBUTION 



ooe and only one solution. But we have found one solution of the problem 

 of equilibrium of energy in a system of material points in motion. If, therefore, 

 the real material system in which the equilibrium of temperature takes place 

 M phLt of being accurately represented by a system of material points (as 

 hfifyj i n pure dynamics) acting on each other according to determinate, though 

 unknown, laws, then the mathematical condition of the equilibrium of energy 

 nmt be the dynamical representative of the physical condition of the equality 

 of temperature. 



It appears from the theorem that in the ultimate state of the system the 



avenge kinetic energy of two given portions of the system must be in the 



of the number of degrees of freedom of those portions. This, therefore, 



must be the condition of the equality of temperature of the two portions of the 



. - 



H.nce at a given temperature the total kinetic energy of a material system 

 must be the product of the number of degrees of freedom of that system 

 into a constant which is the same for all substances at that temperature, being 

 in feet the temperature on the thermodynamic scale multiplied by an absolute 

 constant 



If the temperature, therefore, is raised by unity, the kinetic energy is 

 increased by the product of the number of degrees of freedom into the absolute 

 constant. 



The observed specific heat of the body, expressed in dynamical measure, is 

 the increment of the total energy when the temperature is increased by unity. 

 The observed specific heat cannot therefore be less than the product of the 

 number of degrees of freedom into the absolute constant, unless the potential 

 energy diminishes as the temperature rises. 



Dynamical Specification of the motion. 



\\ B shall begin by supposing the material system to be of the most general 



having ite configuration determined by the n variables q,, q t ...q n , and its 



motion determined by the corresponding momenta p v p t ...p n . The state of the 



system at any instant is completely defined if we know the values of these 



variables for that instant. 



We shall suppose the forces acting between the parts of the system to 

 be of the most general kind consistent with the conservation of energy. This 



