446 MATHEMATICAL APPENDICES 46* 



The numerical factor, which is the same for all gases, does not 

 materially differ from that which has been calculated on the 

 assumption of equality of speed for all molecules. 



47*. Calculation from Maxwell's General Law 



Even this improved calculation cannot lay claim to perfect 

 accuracy. Instead of the law of distribution which is valid for the 

 state of rest, must be employed the law that holds for a gas in a 

 state of flow. 



Maxwell's law can of course be extended without difficulty 

 to the case wherein the mass of gas has a forward motion, as has 

 been shown in 16* and 18*. But this generalisation has 

 reference only to the particular case wherein the forward motion 

 has the same direction and speed at all points in the whole mass 

 of gas. Those formulae, therefore, are not directly applicable to 

 our case, in which the gas is divided into layers which slide past 

 each other with unequally quick motion. 



The formulae can, however, be taken with sufficient approxima- 

 tion as applicable, if their use is limited to so small a region that 

 within it the forward motion of the gas may be considered as 

 everywhere the same in magnitude and direction. Now the 

 transference of momentum, in which, according to the kinetic 

 theory, the process of internal friction consists, is carried on by a 

 molecule no further than the molecule itself moves : it depends, 

 then, on processes in such limited spaces that their dimensions 

 may be compared with the mean length of the molecular free 

 paths. And within such spaces we may look on the speed of the 

 flow as uniform. 



This assumption may be justified quite independently of the 

 hypotheses of the kinetic theory, and simply by means of the 

 assumptions upon which Newton based the theory of viscosity. 

 The validity of the formulae of this theory depends on the 

 limitation that the velocities of the flow in separate layers which 

 slide past each other are taken to be only very slightly different ; the 

 difference in the motion of neighbouring layers is to be taken as 

 so small that, in comparison with the first power of the difference 

 of the velocities, all higher powers may be neglected as vanish- 

 ingly small. 



On these grounds we here employ the formulae of 16* and 



