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kind of free electrons, having all the same charge e and the same 
mass 1; the number of these particles per unit volume will be 
represented by MN, and I shall suppose their heat-motion to have 
such velocities that, at a definite temperature, the mean kinetic 
energy of an electron is equal to that of a molecule of a gas. Deno- 
ting by 7 the absolute temperature, I shall write for this mean 
kinetic energy a7’, where a is a constant. 
We shall further consider a cylindrical bar, unequally heated in 
its different parts, so that, if « is reckoned along its length, 7’ is 
a function of this coordinate. We shall also suppose each electron to 
be acted on, in the direction of OY, by a force mX, whose intensity 
is a function of x. Such a force may be due either to an electric 
field or, in the case of a non-homogeneous metal, to a molecular 
attraction exerted by the atoms of the metal. Our first purpose will 
be to caleulate the number of electrons » and the amount of energy 
W crossing an element of surface perpendicular to the axis of w in 
the positive direction, or rather the difference between the numbers 
of particles in one case and the quantities of energy in the other 
that travel towards the positive and towards the negative side. Both 
quantities » and JI’ will be referred to unit area and unit time. 
This problem is very similar to those which occur in the kinetic 
theory of gases and, just like these, can only be solved in a rigourous 
way by the statistical method of Maxwerr and BorTZMANN. 
In forming our fundamental equation, we shall not confine ourselves 
to the cylindric bar, but take a somewhat wider view of the subject. 
At the same time, we shall introduce a simplification, by which it 
becomes possible to go further in this theory of a swarm of electrons 
than in that of a system of molecules. It relates to the encounters 
experienced by the particles and limiting the lengths of their free 
paths. Of course, in the theory of gases we have to do with the 
mutual encounters between the molecules. In the present case, on 
the contrary, we shall suppose the collisions with the metallic atoms 
to preponderate; the number of these encounters will be taken so 
far to exceed that of the collisions between electrons mutually, that 
these latter may be altogether neglected. Moreover, in calculating the 
effect of an impact, we shall treat both the atoms and the electrons 
as perfectly rigid elastic spheres, and we shall suppose the atoms 
to be immovable. Of course, these assumptions depart more or less 
from reality; I believe however that we may safely assume the 
general character of the phenomena not to be affected by them. 
§ 2. Let dS be an element of volume at the point (w, y, 2). At 
