50 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



Now if we put through the origin O another system of co-ordi- 

 nates of which one axis coincides with any arbitrarily chosen 

 velocity 4 o, then in this axis the above-found product will be 

 nf (p)f(O)-f(O)dV because the two other co-ordinates (outside 

 of p) of the volume element dV will equal zero and no preference 

 can be given to any direction. Then it can be shown that the 

 form of the function is given by 



_i! 



Ce ", ........ (3) 



where C, c are integration constants which stand in a certain 

 relation to each other, namely, 



(4) 



Further mathematical manipulation eliminates the different 

 velocity directions and gives 



, ... (5) 



for the number of molecules possessing absolute velocities between 

 p and p+dp. 



This expression (5) is called MAXWELL'S Law of Distribution ; 

 it is identical with that found for the probable distribution of 

 error in a great number of observations and is graphically shown 

 by the following figure, with maximum number of molecules for 

 velocity c. The constant c is therefore a velocity from which 



curring is equal to nf()d -f(i))d . Similarly, the number of molecules concurrently 

 possessing velocities parallel to each of the three axes is 



The problem is simpler in this Maxwellian case than in the more general case of 

 any state of the body in which there is an unequal distribution in space of the 

 molecules. 



