812 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
And from equation (87) 
p i=u n —\ 
s\ — s'r, da dp~\ 
(91) 
The coe fficient of friction at the solid surface. 
98. Since f or \ is important as regards that which is to follow, it is necessary to 
determine, as far as possible, on what these factors depend. I am not aware that any 
very definite idea has hitherto been arrived at as to the action between the molecules 
of a gas and a solid surface over which the gas may be in motion. It appears to have 
been thought sufficient in most cases to assume that the gas in immediate contact 
with the surface is at rest, which supposition is equivalent to neglecting any small 
motion there may be. 
We see at once that the gas at the surface must have a velocity when the gas 
further away is in motion. For by our fundamental assumption the molecules which 
approach the surface will partake of the motion further away; so that even supposing 
the surface to be perfectly rough, the entire group, consisting of the approaching and 
receding molecules, would have a velocity equal to half that of the approaching 
molecules. 
If the surface be less than perfectly rough, we have, as in equation (89), 
u. 2 =fu\ 
where f~ l may be considered to be the coefficient of roughness. 
Since we have nothing in nature analogous to perfect roughness, we may assume 
that f is not zero, and the question arises whether f may not largely depend on the 
angle at which the molecules approach the plane. 
Even if the solid surface were a perfectly even plane, - would not be the simple 
coefficient of friction, but must also be a function of the force with which the molecules 
strike the surface, and the more nearly perpendicular to the surface was the direction 
of approach the smaller would he the value of f 
Whereas if, as seems highly probable, the action between the molecules and the 
surface is closely analogous to that between a ball and an uneven but perfectly smooth 
elastic surface, then for molecules approaching the surface at very small angles f would 
be unity, while for those approaching in a manner nearly perpendicular / would be 
zero, or nearly so. 
The variation of f with the angle of approach can be of no particular moment so 
long as there is a sufficient thickness of gas between the surface considered and any 
surface which may be opposite, for in that case the mean angle of approach must be 
the same, whatever may be the condition of the gas. But when the gas is between 
two surfaces, as in a tube, and these surfaces are so near that the molecules range 
across the interval, then the fact that if small the angle of reflection (measured from 
