Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/480

This page has been proofread, but needs to be validated.

460

The Theory of Aether and Electrons in the

number added to the group by collisions in the same interval is $b dx du dv dw dt$ . Then we have

$f (u, v, w, x) + (b - a) dt = f (u + eEdt / m, v, w, x + udt)$ ,

and therefore

$b-a={\frac {eE}{m}}{\frac {\partial f}{\partial u}}+u{\frac {\partial f}{\partial x}}$ .

Now, the law of distribution of velocities which Maxwell postulated for the molecules of a perfect gas at rest is expressed by the equation

$f=\pi ^{-{\frac {3}{2}}}a^{-3}Ne^{-{\frac {r^{2}}{a^{2}}}}$ ,

where $N$ denotes the number of moving corpuscles in unit volume, $r$ denotes the resultant velocity of a corpuscle (so that $r 2 = u 2 + v 2 + w 2)$ , and $a$ denotes a constant which specifies the average intensity of agitation, and consequently the temperature. It is assumed that the law of distribution of velocities among the electrons in a metal is nearly of this form; but a term must be added in order to represent the general drifting of the electrons parallel to the axis of $x$ . The simplest assumption that can be made regarding this term is that it is of the form

 $u \times a function of r only$ ;

we shall, therefore, write

$f=N\pi ^{-{\frac {3}{2}}}a^{-3}e^{-{\frac {r^{2}}{a^{2}}}}+u\chi (r)$ .

The value of $χ (r)$ may now be determined from the equation

$b-a={\frac {eE}{m}}{\frac {\partial f}{\partial u}}+u{\frac {\partial f}{\partial x}}$ ;

for on the left-hand side, the Maxwellian term

$\pi ^{-{\frac {3}{2}}}a^{-3}Ne^{-{\frac {r^{2}}{a^{2}}}}$

would give a zero result, since $b$ is equal to $a$ in Maxwell's system; thus $b - a$ must depend solely on the term $uχ (r)$ ; and