1352 
§ 12. In what precedes we were concerned with the volumes of 
parallelepipeds expressed in natural units. When we have intro- 
duced coordinates z,,.. 7, we may also express these volumes in 
the “r-units” corresponding to the coordinates chosen. 
Let us consider e.g. the three-dimensional extension «, = const, 
which cuts the conjugate indicatrix in the ellipsoid 
Inti + Joa®s” + Jaaa + 291,010, + 29g9%2%, + 29,,%,%, = TE. 
If we agree that in z-measure spaces in this extension will be 
represented by positive numbers and that a parallelepiped with the 
positive edges dx,, dr,, de, will have the volume dz, de, dr,, we 
find for that of the parallelepiped on three conjugate radius-vectors 
ra 
Vg 
ia 
where it has been taken into consideration that G,, is negative. 
The volume of the same parallelepiped being expressed in natural 
measure by — Ze’ ($ 8), we have to multiply by 
ea Ge ce OR eet ee 
128. 
if we want to pass from the expression in «-measure to that in 
natural measure. : 
For the extension (z,, 2, 2,), i.e. 2, =O the corresponding factor is 
i ee VS Zeine oe 
§ 13. In the theory of electromagnetic phenomena we are con- 
cerned in the first place with the electric charge and the convection 
current. So far as these quantities belong to a definite element d2 
of the field-figure they may be combined into 
qd 
where q is a vector which we may call the current vector. When 
it is resolved into four components having the directions of the axes, 
the first three components determine the convection current, while 
the fourth component gives the density of the electric charge. 
As to the electric and the magnetic force, these two taken together 
can be represented at each point of the field-figure by two rotations 
R. and ‘R; fe 
in definite, mutually conjugate two-dimensional extensions. These 
quantities are closely connected with the current vector, for after 
having introduced coordinates 2,,...2, we have for each closed 
surface o the vector equation 
. 
ri. 
