as one can easily verify by substituting from (30a) and (31). Equations (32) and (33) have a tensor character, and are therefore co-variant with respect to Lorentz transformations, if the and the have a tensor character, which we assume. Consequently, the laws for transforming these quantities from one to another allowable (inertial) system of co-ordinates are uniquely determined. The progress in method which electrodynamics owes to the theory of special relativity lies principally in this, that the number of independent hypotheses is diminished. If we consider, for example, equations (19a) only from the standpoint of relativity of direction, as we have done above, we see that they have three logically independent terms. The way in which the electric intensity enters these equations appears to be wholly independent of the way in which the magnetic intensity enters them; it would not be surprising if instead of , we had, say, or if this term were absent. On the other hand, only two independent terms appear in equation (32). The electromagnetic field appears as a formal unit; the way in which the electric field enters this equation is determined by the way in which the magnetic field enters it. Besides the electromagnetic field, only the electric current density appears as an independent entity. This advance in method arises from the fact that the [Pg 43] electric and magnetic fields draw their separate existences from the relativity of motion. A field which appears to be purely an electric field, judged from one system, has also magnetic field components when judged from another inertial system. When applied to an electromagnetic field, the general law of transformation furnishes, for the special case of the special Lorentz transformation, the equations [10]In order to avoid confusion from now on we shall use the three-dimensional space indices, , , instead of 1, 2, 3, and we shall reserve the numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum. If there exists with respect to only a magnetic field, , but no electric field, , then with respect to ' there exists an electric field as well, which would act upon an electric particle at rest relatively to '. An observer at rest relatively to would designate this force as the Biot-Savart force, or the Lorentz electromotive force. It therefore appears as if this electromotive force had become fused with the electric field intensity into a single entity. In order to view this relation formally, let us consider the expression for the force acting upon unit volume of electricity, in which is the vector velocity of electricity, with the velocity of light as the unit. If we introduce and according to (30a) and (31), we obtain for the first component the expression [Pg 44] Observing that vanishes on account of the skew-symmetry of the tensor (), the components of are given by the first three components of the four-dimensional vector