58 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



whose edges are parallel to the co-ordinate axes ; this is the simplest 

 of the elementary regions here to be considered. To conceive 

 of the elementary region ds-dy-d"^ containing the velocity ends of 

 the molecules, let us suppose any origin O for velocities in a unit 

 volume and from this as a pole lay off as vectors the molecular 

 velocities lying between the limits, 



and 



and y+dr), ...... (n) 



and 



where , y, are the components of said velocities parallel to the 

 respective co-ordinate axes. Then, under the velocity limitations 

 imposed, the end of the velocity of each such molecule will lie in 

 the elementary parallelepiped d-dr}-d^, one vertex of this 

 parallelepiped having of course the co-ordinates , y, . This 

 parallelepiped can be regarded as a constructed volume within 

 which the velocity end must lie. We have therefore here two 

 elementary volumes dx-dy- dz and df-dy-dl, which are independent 

 of each other. Now remembering that the probability of any 

 properly endowed molecule being found in one of these volumes 

 is in each case equal to the number of molecules belonging or 

 corresponding to the volume considered. Assuming, for the 

 moment, an equal distribution of molecules and velocities through- 

 out the whole volume, we may say that the number of molecules 

 occurring, in each of the said elementary volumes, is proportional 

 to their respective sizes; this is here equivalent to saying that 

 the probability of any molecule thus occurring in said elementary 

 volumes is proportional to their respective sizes. Having stated 

 the probability of each contemplated occurrence, we can now say 

 the probability of these two events concurring is equal to the product 

 of the probabilities of said two separate occurrences. Moreover, 

 as the probability of each occurrence is proportional to the size 



