380 Mr. W. Sutherland on Thermal 



slipping, and shows how slipping arises, but does not give its 

 amount correctly ; this, however, can soon be obtained. Let 

 iu 1 be the sudden change of velocity on passing from solid to 

 gas ; then the Xdiv/dx4: just given must be increased by w 1} and 

 then ihe average loss of momentum experienced by a molecule 

 encountering the fixed surface is m^fXdw/dxi + w^ ; but the 

 number encountering unit surface in unit time is (2) nv/4:, 

 and therefore the frictional force exerted by unit surface of 

 solid on the gas is nmv(/Xdw/dx4: + t0 1 )/4, which is equal to 

 ydw/dx, the friction on parallel unit surface in the gas when 

 the motion is steady : thus 



dw(lr L _fX 

 ax\nmv 



); (8) 



but r) = '365nmv\ or, working with the same methods of 

 approximation as we have been using, rj = nmvX/4: } and then 



«*i = ^Ml-//4) (9) 



/ is a fraction which from its nature is unlikely to exceed 

 1/2, so that we can write iv } =aXdw/dx with the knowledge 

 that a is not much different from unity. At both the moving 

 and the fixed surfaces there is this discontinuity of amount 

 Wi, so that in the theory of viscosity, instead of writing 

 dw/dx = tv/T> for the steady state, we must write 



dw _ id — 2a\ % div _ iv 



dx"~ ~ D~ ' ''• fa~ D{l + 2a\/D) ' ( ' 



aX or f is called the coefficient of slipping ; under ordinary 

 circumstances it may be neglected, but when D is comparable 

 with X, as it mostly is in connexion with thermal transpiration 

 and radiometer motion, slipping becomes of fundamental 

 importance. When D is only a fraction of X viscosity 

 practically ceases, because the molecules traffic backwards 

 and forwards between the solids with so few encounters 

 amongst themselves that they hardly affect one another's 

 motion, but they still exercise friction on the solids whose 

 amount is easily calculated. Suppose that the gas between 

 two parallel solid planes at rest is also at rest, except of course 

 for the velocities of agitation, and then let one of the planes 

 be set moving parallel to itself with velocity iv; then, as we 

 have seen, the molecules colliding with it leave on the average 

 with velocity fw, and when they reach the fixed plane a 

 fraction / will have this velocity reduced to zero, while 1 — f 

 will retain it unaltered, so that on the average the molecules 



