and the Partition of Energy in Continuous Media. 235- 



concentrate about any special points or regions in the 

 generalized space, or the reverse. 



We have so far merely studied the changes in the values 

 of a system of algebraic variables, when they change as 

 directed by a certain algebraic system o£ conditions (namely,. 

 $(ljdt = 0). The motion of points in the generalized space 

 has merely provided a graphical representation of these 

 changes. 



9. Now let these variables # 1? # 2 , . . . 6 n , # l5 q , . . . 6 n be the 

 coordinates and velocities of a dynamical system, and let L 

 be its Lagrangian function. Then the motion of a " repre- 

 sentative point " in the generalized space will represent the 

 changes in the coordinates and velocities as these change in 

 accordance with the principle of Least Action — i. e., as the 

 system moves in accordance with the laws of nature. The 

 proved fact that in this motion there is no tendency for the 

 " representative points " to concentrate about any special 

 points or regions of the generalized space leads at once to the 

 following : — 



Theorem. All properties (if any) which are such as to be 

 finally acquired by the dynamical system, independently of the 

 special state from which the system started, must be properties 

 common to the ivJiole of the generalized space. 



For, if the system must inevitably possess some property, 

 this can only be either because its representative point tends 

 inevitably to pass into the regions of the generalized space in 

 which this property holds, or else because the property holds 

 in all regions. The former alternative is disproved by 

 equation (7) : the latter alternative must accordingly be the 

 true one. 



10. It is found that, in general, there are no properties 

 common to all regions iti the generalized space (or rather, no 

 properties of any importance for the present purpose). But 

 when the system possesses an infinite (or very great) number 

 of similar coordinates, there are certain statistical properties 

 found to be common to the whole of the space except for 

 infinitesimal regions of it. A system which possesses these 

 statistical properties is said to be in the " Normal State." 



A representative point may of course have its whole path 

 in regions in which the " Normal State" does not obtain, or 

 it may pass through these regions for periods, large or small, 

 on its path. But we have the quite general theorem: — 



Theorem. If a system tends to acquire definite properties, 

 independently of its initial configuration, or if it tends to acquire 



