PROPERTIES OF MATTER IN THE GASEOUS STATE. 
789 
2 . That s for the mean cube does not enter into any of the experimental results of 
this investigation; 
3. That s for the mean velocity has a real value, but there are no data for effecting 
a numerical comparison between this and the other value of s. 
As this foundation of the theory on elementary assumptions renders it more satis¬ 
factory, it is introduced at length into this section of the paper. The argument, which 
is long and occupies Arts. (79 to 84), may be sketched as follows :— 
Sketch of the method by which the fundamental theorems are deduced. 
76. Upon certain elementary assumptions, which do not involve any particular law of 
action between the molecules, it is first shown that, in respect of density, mean velocity, 
&c., considered separately, any group of molecules whose directions of approach differ 
by less than a given small angle from any given direction BA, will enter the element 
at A (within a sufficient degree of approximation) as if the gas were uniform and had the 
same density and mean pressure as at B, and had mean component velocities which, 
although not the mean component velocities at B, are equal to one half the mean 
component velocities of all the molecules which enter an indefinitely small element at 
B in a unit of time. These component velocities, which are written U, V, W, cannot 
in the first instance be expressed in terms of known quantities, but they are shown to 
be functions of the position of B in the gas. 
The distance AB or s is shown to be a function of the pressure and density of the 
gas, which function, although not completely expressed, as such an expression would 
involve the law of action between the molecules, is shown to be approximately inde¬ 
pendent of the variation of the density and pressure, and hence of the direction of AB. 
The relations between p, a, U, V, W for a uniform gas may thus be used to express 
severally the density, mean velocity, &c., for each elementary group of molecules 
arriving at A. And since p, a, TJ, V, W are functions of the position of the point B (if 
x y z are the coordinates of A, and l m n are the direction cosines of AB) they are 
functions of x-\-ls, y-\-ms, z-{-ns, s having the value for the particular quantity to be 
represented. Therefore p , a, U, Y, W for B may, by expansion, be represented by p, a, 
U, Y, W for A, and their differential coefficients multiplied by powers of s. Thus the 
density, mean velocity, &c., of the molecules of each group arriving at A may severally 
be expressed in terms of p, a, U, Y, W at A, and their differential coefficients 
multiplied by a particular value of s. 
Therefore as the elementary portions of cr(Q) for the group can always be expressed 
in terms of the density, mean velocity, &c., and /, m, n, it can be expressed in terms 
of p, a, U, Y, \Y, for A, their differential coefficients multiplied by certain values of s 
and l, m, n. And, since all these quantities but l, m, n are independent of the 
direction of the group, by integrating for all values of l, m, n, cr(Q) is found in terms 
of p, a, U, Y, W, for A, and their differential coefficients multiplied by s. 
