364 MATHEMATICAL APPENDICES 7* 



since %7 = ; this is greater than the other components by pa 2 , 

 for we also have 



P V = 3P w i 2 - 



This new magnitude pa 2 shows itself as the pressure with which 

 the streaming gas strikes against an opposing surface, or, if we 

 consider the reverse direction, the reacting pressure which the 

 issuing stream exerts on the containing vessel ( 34, 35). The 

 pressure which the stream of gas exerts normally to its direction 

 of flow, viz. 



is smaller than what it would be in the case of rest, viz. 



for in coming to rest the gas retains all its energy, so that the 

 mean value of this energy remains unaltered, or 



The formula found for the pressure in the case of a gas at 



rest, 



does not differ from that deduced before ( 11-13), viz. 



since the magnitude G herein contained denotes the mean value 

 which is determined by the equation 



Since this formula gives the pressure p as proportional to the 

 density p, it expresses Boyle's law. 



This deduction of the law is preferable to the former in being 

 as free as possible from unproved hypotheses. We have assumed 

 only that the molecules move in rectilinear paths, and have 

 employed no other hypothesis. The single assumption, therefore, 

 of rectilinear molecular motion suffices by itself for the proof of 

 Boyle's law. 



Since the sum of the kinetic energy in the unit of volume is 



K = itfmf du\ & dw (u* + v z + w*)F(u, v, w) 



J 00 J CO J CO 



