Page:BatemanElectrodynamical.djvu/7

and this is equivalent to

$-\iiint \left(\rho w_{x}{\frac {\partial t}{\partial x}}+\rho w_{y}{\frac {\partial t}{\partial y}}+\rho w_{z}{\frac {\partial t}{\partial z}}-\rho \right)dx\ dy\ dz,$

if equations (I) are satisfied and the axes form a right-handed system. In the same way, it can be shown that the integral on the left-hand side of (II) is zero, if equations (I) are satisfied.

Conversely, if equations (II) and (III) are satisfied for every closed surface S and for every law $t=t(x,y,z)$, such that $\partial t/\partial x,\ \partial t/\partial y,\ \partial t/\partial z$ are finite and continuous within S and on its surface, then equations (I) are also satisfied, provided, of course, that the quantities $E_{x},\dots ,H_{x,}\dots$ possess derivatives which behave in such a way that an application of Green's theorem is permissible.

It seems natural to regard equations (II) and (III) as the fundamental equations of the theory of electrons, because they do not require the assumptions that the medium is continuous and that vectors E, H can be associated with every point in space;^[1] all that is required is that the quantities $E_{x},\dots ,\rho w_{x,}\dots$ shall be integrable.

It is clear that in the general case the quantities occurring in these equations are evaluated at different points of space at different times. The integrals are thus more general than the usual surface and volume integrals, and seem to be better adapted for purposes of measurement, the difficulty of measuring quantities at different points of space at the same time being avoided.

Equation (III) may be regarded as the definition of the electric charge associated with a system of particles. The triple integral represents the total charge on the particles.^[2] Each particle is supposed to be within

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1. In the case of a single pulse travelling across a medium in which the vectors E and H are initially zero, these vectors may exist at a given point at one time and not at another.
2. This may be proved by showing that the triple integral remains invariant during the motion of a system of electrons. Let an electron which was at the point (x, y, z) at time t be in a new position (x', y', z') at time t', where

   ${\begin{array}{lll}x'=x+\epsilon w_{x},&&z'=z+\epsilon w_{z},\\\\y'=y+\epsilon w_{y},&&t'=t+\epsilon ,\end{array}}$

   and t is a small quantity which is a function of x, y, z, t. The integral form is transformed into

   $\rho 'w'_{x}dy'dz'dt'+\rho 'w'_{y}dz'dx'dt'+\rho 'w'_{z}dx'dy'dt'-\rho 'dx'dy'dz',$

   where

   $\rho w_{x}=\rho 'w'_{x}{\frac {\partial (y',z',t')}{\partial (y,z,t)}}+\rho 'w'_{y}{\frac {\partial (z',x',t')}{\partial (y,z,t)}}+\rho 'w'_{z}{\frac {\partial (x',y',t')}{\partial (y,z,t)}}-\rho '{\frac {\partial (x',y',z')}{\partial (y,z,t)}},$

   $\dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots$

   or

   $\rho 'w'_{x}{\frac {\partial (x',y',z',t')}{\partial (x,y,z,t)}}=\rho w_{x}{\frac {dx'}{dx}}+\rho w_{y}{\frac {dx'}{dy}}+\rho w_{z}{\frac {dx'}{dz}}+\rho {\frac {dx'}{dt}}.$

   $\dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots$

   This gives

   ${\begin{array}{ll}\rho 'w'_{x}{\frac {\partial (x',y',z',t')}{\partial (x,y,z,t)}}=&\rho w_{x}\left[1+\epsilon {\frac {\partial w_{x}}{\partial x}}+w_{x}{\frac {d\epsilon }{\partial x}}\right]+\rho w_{y}\left[\epsilon {\frac {\partial w_{x}}{\partial y}}+w_{x}{\frac {d\epsilon }{\partial y}}\right]\\\\&+\rho w_{z}\left[\epsilon {\frac {\partial w_{x}}{\partial z}}+w_{x}{\frac {d\epsilon }{\partial z}}\right]+\rho \left[\epsilon {\frac {\partial w_{x}}{\partial t}}+w_{x}{\frac {d\epsilon }{\partial t}}\right]\end{array}}$

   $\dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots$

   $\rho '{\frac {\partial (x',y',z',t')}{\partial (x,y,z,t)}}=\rho w_{x}{\frac {d\epsilon }{\partial x}}+\rho w_{y}{\frac {d\epsilon }{\partial y}}+\rho w_{z}{\frac {d\epsilon }{\partial z}}+\rho \left[1+{\frac {\partial \epsilon }{\partial t}}\right].$

   Multiplying the last equation by $w_{x}$, and subtracting from the first, we get

   $\rho '{\frac {\partial (x',y',z',t')}{\partial (x,y,z,t)}}\left(w'_{x}-w_{x}\right)=\rho \epsilon \left[{\frac {\partial w_{x}}{\partial t}}+w_{x}{\frac {\partial w_{x}}{\partial x}}+w_{y}{\frac {\partial w_{x}}{\partial y}}+w_{z}{\frac {\partial w_{x}}{\partial z}}\right].$

   Putting

   ${\frac {d}{dt}}={\frac {\partial }{\partial t}}+w_{x}{\frac {\partial }{\partial x}}+w_{y}{\frac {\partial }{\partial y}}+w_{z}{\frac {\partial }{\partial z}},$

   we see that the equation is satisfied if

   $\left(1+{\frac {d\epsilon }{dt}}\right)\left(w'_{x}-w_{x}\right)=\epsilon {\frac {dw_{x}}{dt}},$

   and if ${\frac {d\epsilon }{dt}}$ is a small quantity of the first order

   $w'_{x}-w_{x}=\epsilon {\frac {dw_{x}}{dt}},$

   in other words $\left(w'_{x},w'_{y},w'_{z}\right)$ are the component velocities at time t + ε.

   Since the integral form is an invariant, we may calculate its value by considering the swarm of electrons at a given time t. The temporal terms then disappear, and the integral reduces simply to

   $-\rho \ dx\ dy\ dz$

   i.e., the total negative charge on the system of electrons.