78 G 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
Section VII.— The Mean Range. 
75. So far the gas has been supposed to be in uniform condition as regards space as 
well as time. When the condition varies from point to point, the results given in Table 
XX. will not hold good, for the condition of the molecules arriving from any particular 
direction which cross a plane at a point A will not be determined by the mean con¬ 
dition of the gas at A, but rather by the mean condition at the points at which the 
molecules receive the direction and velocity with which they cross the plane. 
These points will not necessarily be the points at which the molecules last undergo 
encounter before crossing the plane, for one encounter may not be sufficient completely 
to modify their motion. In order, therefore, to determine from first principles the 
manner in which the molecules approach the point A, we must know the law of action 
between the molecules, and even then the complete solution would present difficulties 
which appear to be insuperable. 
Fortunately, however, for the purposes of the present investigation a complete 
solution is not necessary. The point that has mainly to be considered is the effect of a 
solid surface on the mean condition of the molecules which cross a plane in its im¬ 
mediate neighbourhood. And the principal question is not how far such an effect 
would extend into gas in a particular condition, but what would be the nature of the 
effect at points to which it does extend, and what would be the comparative range 
of similar effects in gases the condition of which differ with respect to density and 
variation of temperature ? If it should be found that the number and mean condition 
of the molecules which arrive at A from a given direction partake in a definite manner 
of the condition of the gas at a point in that direction whose distance s from A is a 
definite function of the density of the gas and some function of the temperature ; 
such a solution would be sufficient to allow of the deduction of results corresponding 
to the experimental results. 
Now it appears to follow from the view propounded at the commencement of this 
article, that in the interior of the gas there must be some distance s from a point A at 
which the mean condition of the gas must represent the mean condition of the mole¬ 
cules which reach A from that direction. This language is somewhat vague, but so 
must be the first idea. On closer inspection the question naturally arises as to what 
is meant by the mean condition of the gas, and by the mean condition of the molecules 
which reach A ? Nor does this question at first sight appear to be difficult to answer. 
The mean condition of the gas appears most naturally to resolve itself into that 
which we can measure—the density p, the mean pressure ~ (w 2 +ffi+tt> 2 ) and the 
O 
mean component velocities u, v, w ; and with respect to the mean condition of the 
arriving molecules why should not this be measured by their density, their mean 
energy and their mean component velocities ? On comparing these with the cor¬ 
responding quantities for the gas just mentioned, one point of doubt presents itself: 
