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PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
(1) It is not implied that the mean range is the same for all the quantities which 
may be considered; 
(2) There is no fear of confusing the mean range with the mean path of a molecule. 
The mean range, whatever may be the nature of the quantity considered, is obviously 
a function of the mean path of the molecules, and is a small quantity of the same 
order as the mean path, but it also depends on the nature of the impacts between the 
molecules. 
The symbol s is used to express the mean range of any particular quantity Q. 
62. Assuming that the mean value of Q for the molecules in an elementary unit of 
volume at a point is a function of the position of the point, the aggregate value of Q 
carried across the plane at a point is obtained in a series of ascending powers of s. 
And by neglecting the terms which involve the higher powers of s, which terms also 
involve differentials of Q of orders and degrees higher than the first, equations are 
obtained between s and the aggregate value of Q carried across the plane. 
63. The dynamical conditions of steady momentum, steady density, and steady 
pressure are next considered. General equations are obtained for these conditions, 
which general equations involve s, the motion of the plane and other quantities 
depending on the condition of the gas. 
The condition that there may be no tangential stress in the gas is also considered. 
It is found that when there is no tangential stress on a solid surface wherever it 
may be in the gas, the mean component velocities of all the molecules which pass 
through the element in a definite time must be zero at all points in the gas. 
64. The equations of motion are then applied to the particular cases which it is the 
object of this investigation to explain. Two cases are considered. The first, that of a 
gas in which the temperature and pressure only vary along one particular direction, so 
that the isothermal surfaces and surfaces of equal pressure are parallel planes ; this 
is the case of transpiration. The second case is that in which the isothermal surfaces 
and the surfaces of equal pressure are curved surfaces (whether of single or double 
curvature); this is the case of impulsion and the radiometer. 
As regards the first case, the condition of steady pressure proves to be of no 
importance; but from the conditions of steady momentum and steady density an 
equation is obtained between the velocity of the gas, the rate at which the tem¬ 
perature varies, and the rate at which the pressure varies; the coefficients being 
functions of the absolute temperature of the gas, the diameters of the apertures, and 
the ratio of the diameters of the apertures to the mean range. These coefficients are 
determined in the limiting conditions of the gas, when the density is small and large, 
and as they vary continuously with the condition of the gas, the limiting values afford 
indications of what must be the intermediate values. 
From this equation, which is the general equation of transpiration, the experimental 
results, both as regards thermal transpiration and transpiration under pressure, are 
deduced. 
