852 
v4, take an element of volume dV, situated at the point 2, 2®), 
xv), It will contain NdV particles. We assume that NdV is still 
a great number, notwithstanding d@V being physically infinitesimal. 
Now, in our four-dimensional picture, consider the trails of these 
NdV particles, run through during an interval of time de). These 
trails will cover an element of space-time-extension of magnitude 
dVda@). In the direction of the coordinate X* the components will 
in the aggregate amount to 
Nd Vda. 
It will readily be seen that the streamcomponent in the direction 
of Xe is the aggregate of the X*-components of the four-dimensional 
trails, run through by the particles per unit of volume per unit of time: 
NdV dae dat N DA 
= N= = Nue == il, 4, 8,4) 
dV da =e dr or Ue (a = Sh So So) ) 
We shall put wd, w®, w) for the components of the velocity: 
de da, dede, da)/da. The fourth component equals unity: 
wt) = de/dx®, and accordingly the fourth streamcomponent Nw) 
is the number of particles per unit of volume. 
It is obvious that the equation of continuity must be satisfied by 
these streamcomponents : 
__, ONw! 
= (b) Sb 0 
By the displacements the components will change to 
Nwt + dNwe + 4d? Nwt, 
where the first variation 0 Nw is proportional to 6 and the second 
variation J? Nw will contain the second order terms with 67. It 
may be anticipated that the first variation will account for by 
far the greater part of the effects of polarization, whereas the second 
variation mainly gives the effects of magnetization. 
4. We proceed to the evaluation of the first variation. Here we 
may consistently neglect 6’. 
The displacements will have changed the aggregate of the Xc- 
components of the trails under consideration: each dx* passes into 
OO F 
ae vb, 
dat 4 = (b) 
so that the aggregate becomes 
N dV {dae + & (b) dx? 
dgre 
Ox 
On the other hand the four-dimensional extension covered by the 
