Transpiration and Radiometer Motion. 379 



Kundt and Warburg (Pogg. Ann. clvi.) showed experi- 

 mentally the existence of slipping by its effect on the apparent 

 coefficient of viscosity at low enough densities of the gas in 

 an oscillating disk apparatus for measuring viscosity, and 

 they adduced theoretical reasons for the necessity of its 

 existence and for some of its properties; they also measured 

 its amount and verified some of its laws, and a little later 

 Warburg demonstrated the slipping of gas on the walls of 

 capillary tubes (Pogg. Ann. clix.) 



That slipping is a necessary consequence of the kinetic 

 theory can easily be shown. Consider gas between two solid 

 parallel planes, one fixed and the other moving parallel to 

 itself with velocity 10; then in the steady state there is a 

 constant rate of diminution of velocity dwjdx in the gas 

 between the plates. Suppose the molecules of the solid, like 

 those of the gas, to be smooth spheres oscillating, but their 

 centres at the surface having a mean position forming a 

 plane. Consider a molecule of gas in collision with a 

 molecule of solid; if its velocity of rebound makes an angle 

 less than 7r/2 with the normal to the plane, the molecule has 

 little chance of colliding with another surface molecule of the 

 solid and is directly reflected; the majority of these directly 

 reflected molecules of gas must strike the molecules of solid 

 near their most prominent points, and therefore acquire from 

 them very little of their velocity parallel to the plane ; thus 

 a certain fraction /of the molecules of gas that encounter the 

 surface leave it with practically the same velocity parallel to 

 it as that with which they approached ; the remaining fraction 

 1— /, or those which at the instant of rebounding from a surface 

 molecule have directions making an angle greater than ir/'2 

 with the normal to the surface, must each penetrate into the 

 hollow between two neighbouring solid molecules and suffer 

 a second encounter with one of them under conditions which 

 necessitate its taking up on the average any motion that the 

 surface has parallel to itself. 



Now suppose that on the average the molecules of gas 

 which collide with the solid come a distance X/2 since their 

 last collision with molecules of gas; then their average 

 distance normally from the surface at the instant of last 

 collision with their fellows will be the average distance of a 

 hemisphere of radius \/2 from its base, which is \/±, and thus 

 the molecules of gas which collide with the solid, which is 

 fixed, reach it with a relative molar velocity \du) ! d,v± ; but 

 after the collision only the fraction f retain this, so that the 

 gas in contact with the solid surface may be said to retain as 

 a whole the velocity fKdw/dx-i, which constitutes a velocity of 



