GASES ARISING FROM INEQUALITIES OF TEMPERATURE. 237 
Let Y (,l) be the value at P of the surface harmonic of order n in the series con¬ 
sidered. 
After the encounter, the corresponding term becomes what Y (i!) becomes at the point 
P', and since all positions of P' in a circle whose centre is P are equally probable, the 
mean value of the function after the encounter must depend on the mean value of the 
spherical harmonic in this circle. 
Now the mean value of a spherical harmonic of order n in a circle, the cosine of whose 
radius is /x, is equal to the value of the harmonic at the pole of the circle multiplied by 
P (il) (p), the zonal harmonic of order n, and amplitude /x. 
Hence, after the encounter, Y (il) becomes Y (re) P ( " ) (/x), and if F n is the corresponding 
part of the function to be considered, and SF„ the increment of F, t arising from the 
encounter, SF w =F n (P (n) (p) —1). 
This is the mean increment of F u arising from an encounter in which cos2d=/x. 
The rate of increment is to be found from this by multiplying it by the number of 
encounters of each molecule per second in which /x lies between /x and yf-dy, and 
integrating for all values of/x from —1 to + 1. 
This operation requires, in general, a knowledge of the law of force between the 
molecules, and also a knowledge of the distribution of velocity among the molecules. 
When, as in the present investigation, we suppose both the molecules to be of the 
same kind, and take both molecules into account in the final summation, the spherical 
harmonics of odd orders will disappear, so that if we restrict our calculations to 
functions of not more than three dimensions, the effect of the encounters will depend 
on harmonics of the second order only, in which case P <:) (y) — 1 =§(/x 3 —l) = f sin 2 2 6. 
—Note added May, 1879.] 
(2.) Number of Encounters in unit of Time. 
We now abandon the dynamical method and adopt the statistical method. Instead 
of tracing the path of a single molecule and determining the effects of each encounter 
on its velocity-components and their combinations, we fix our attention on a particular 
element of volume, and trace the changes in the average values of such combinations 
of components for all the molecules which at a given instant happen to be within it. 
The problem which now presents itself may be stated thus : to determine the dis¬ 
tribution of velocities among the molecules of any element of the medium, the current- 
velocity and the temperature of the medium being given in terms of the coordinates 
and the time. The only case in which this problem has been actually solved is that 
in which the medium has attained to its ultimate state, in which the temperature is 
uniform and there are no currents. 
Denoting by 
hN =f(f, 7], L x, y, z, t)dfhfflxdydz 
