478 Mr. W. Sutherland on Thermal 



(4), and the number that encounter unit surface in unit time 

 being nu/4, the total momentum imparted to the gas in unit 

 time by a surface S is given by 



F = Snwwu/4 (20 a) 



This is the initial value of the traction ; but as the velocity u 

 is carried out to the remoter parts of the gas, a molecule 

 which encounters the surface having come from a region 

 where it had already acquired a fraction of u does not receive 

 the whole of u from the solid, and therefore the traction 

 diminishes with time. To determine the final value when the 

 motion of the whole gas is steady we may consider the simple 

 case of two parallel planes the variations of temperature over 

 which are such as to produce velocities u x and u 2 in a fixed 

 direction in the gas in contact with the two surfaces ; then 

 in the steady state we may suppose the transition from u Y to 

 u 2 to occur linearly, so that the velocity u at distance x from 

 one of the planes is u\ — (ui — u 2 )x/'D J where D is the distance 

 between the planes ; then the mass of gas that flows in unit 

 time along any layer of width b and thickness dx is nmbu dx, 

 and the momentum imparted in unit time to the layer is 

 nmbu 2 dx, and therefore the total momentum acquired by the 

 gas between the planes is 



'D 



nmbvP dx — nmbD (u 2 + u l 7i 2 + u 2 ) /3. 



f 



Obviously the planes impart the respective fractions 



u 



i /(% 2 + w 2 2 ) an d u 2 j{u 2 + u 2 ) of this, so that the traction 

 per unit area of the first plane, if its length in the direction 

 of motion is I, is 



nmDu^fii-j 2 + u^u 2 + ?/ 2 2 ) 

 31(^+^7) ; 



but it is really a useless artificiality to consider the traction 

 per unit surface, as most of the traction is really exerted on 

 the gas near its entrance to the space between the planes, and 

 we will therefore confine our attention to the total tractions. 

 As before in the case of the tube, the result that the traction 

 should be proportional to the sectional area between the plane 

 is peculiar, but it is true only when the planes dominate the 

 temperature of the gas in such a manner that m is a linear 

 function of the distance from either. Thus the initial total 

 traction on the first plane is Snmvu/A, which is proportional 

 to the surface S, that is to both width and length, but inde- 

 pendent of distance from neighbouring surfaces ; and the 

 final traction in the steady state is 



nmhD (it-i 2 + Uiii 2 + u 2 2 )u^l?> (u 2 + u 



