720; REPORT—1895. 
he sad | 
n 
And we can now describe the motion of the medium of finite spheres as follows : 
If it be given that there are 7 spheres within a spherical space 8, but nothing is 
known of their positions within §, the chance that they shall have. velocities 
U, .. Uj tdu,... Wy. . + Wr+dw, is proportional to e~%@du, ... dwn, and 
Q = a3(u? + v* + w) + bE3(uw’ + vv’ + ww’), and 
Cae cama? 
n 
and 
be = provided & be small. 
10. If T be the whole kinetic energy of the 2 molecules, T, the kinetic 
energy of their motion relative to their common centre of inertia, 
Q="+KT., 
and Q is that which is constant in a vertical column under the action of gravity. 
11. The function M which we have used, if we add to it the corresponding 
dé d 
terms in dy 29 &c., can be shown to be the rate of time variation of 
ly ‘ 
dx 
ie k= NG +n? + C)dadydz. 
The investigation shows that H increases or diminishes with the time according as 
ates +7? +?) is below or above the limit = 
2 o 
12. Dr. Ladislas Natanson, in his ‘ Sur |’Interprétation cinétique de la Fonction. 
de Dissipation,’ defines as follows:—Let & 7 ¢ be the component velocities of the 
centre of inertia of the molecules in ‘un élément de volume contenant n.dvdydz 
molécules,’ while x v w are the velocities of a molecule relative to that centre of 
inertia. (I have interchanged Dr. Natanson’s letters.) ‘Then he takes 
H= NG ++ C)dadydz, 
E= {Joe +0 + w*)dadydz 
throughout the space filled by molecules. And he shows that in nature H tends 
to diminish, its energy being converted into the energy, E, of molecular motion. 
_ Dr. Natanson’s definition (though I am not contesting its suffici ney for his 
own purpose) is inapplicable to molecules of finite dimensions, because _a system 
of such molecules ‘ un élément de volume contenant .dvdydz molécules’ does not 
exist. But the function which in this case corresponds to Natauson’s function 
H is 
NG +2 + (?)dxdydz, 
which does, as we have seen, tend to a limit, though not to the limit zero. 
13. The theorem of Arts. 5 and 6, and therefore the whole of this investiga— 
tion, would apply to the case where the molecules, instead of being conventional 
: ; d ..- am 
elastic spheres, are centres of repulsive force, only the exact value of a (V2V.) will 
not be the same. It would still be of the same sign as dg which is the essential 
dz 
characteristic. 
