60 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



Strictly speaking, the expression a for the fictitious elementary 

 region o, formed by the product of dV and the constructed- 

 volume element d dy - d^, should be replaced by the expression 

 w 3 <j, where m is the mass of a molecule. The reason for this 

 substitution is found in the fact that the magnitude of the con- 

 structed- volume element dZ-dy-d^ varies with time due to the 

 variation of velocities effected by molecular collisions. Now 

 this variation of magnitude is not permissible with the probability 

 considerations which here obtain. For the probability of a state 

 which necessarily follows from another state must be like that 

 of the latter. As the momenta after collision are the same as 

 before collision, we have now in the momenta, co-ordinates which 

 do not vary with time like their constituent velocities. Therefore 

 if we substitute in (12) for the velocities , y, their corresponding 

 momenta, the variation with time of the constructed-volume will 

 cease and the objection cited will no longer be a valid one. 



Now let us take up the determination of the number of com- 

 plexions W in the given state. For this purpose think of this 

 whole state as represented by the sum total of all these equal 

 elementary regions m z a\ for convenience of reference let us call 

 this whole state the " state-region." The probability that a par- 

 ticular molecule will belong to a particular elementary region 

 is equally great for all the elementary regions. Let P represent 

 the number of these equal elementary regions. Now we will 

 proceed with the help of a parallel case. Let us think of as many 

 dice N as there are molecules and let each die be provided with 

 P faces. On each of these faces we will write in their order the 

 digits i, 2, 3, ... P, so that each of the P faces will designate 

 a particular elementary region. Then each throw of the N 

 dice will result in representing a particular state of the gas, because 

 the number of dice which show uppermost a particular digit 

 will give the number of molecules belonging to the elementary 

 region represented by said digit. In this parallel case each die is 

 equally likely to show up any one of the digits i to P, corresponding 



