and the fourth component is given by There is, therefore, a four-dimensional vector of force per unit volume, whose first three components, , , , are the ponderomotive force components per unit volume, and whose fourth component is the rate of working of the field per unit volume, multiplied by . A comparison of (36) and (35) shows that the theory of relativity formally unites the ponderomotive force of the electric field, , and the Biot-Savart or Lorentz force [, ]. Mass and Energy. An important conclusion can be drawn from the existence and significance of the 4-vector . Let us imagine a body upon which the electromagnetic field acts for a time. In the symbolic figure (Fig. 2) designates the -axis, and is at the same time a substitute for the three space axes , , ; designates the real time axis. In this diagram a body of finite extent is represented, at a definite time , by the interval the whole space-time existence of the body is represented by a strip whose boundary is everywhere inclined less than 45° to the -axis. Between the time sections, = and = , but not extending to them, a portion of the strip is shaded. This represents the portion of the space-time manifold [Pg 45] in which the electromagnetic field acts upon the body, or upon the electric charges contained in it, the action upon them being transmitted to the body. We shall now consider the changes which take place in the momentum and energy of the body as a result of this action. FIG. 2. We shall assume that the principles of momentum and energy are valid for the body. The change in momentum, , , , and the change in energy, are then given [Pg 46] by the expressions Since the four-dimensional element of volume is an invariant, and (, , , ) forms a 4-vector, the four-dimensional integral extended over the shaded portion transforms as a 4-vector, as does also the integral between the limits and , because the portion of the region which is not shaded contributes nothing to the integral. It follows, therefore, that , , , form a 4-vector. Since the quantities themselves transform in the same way as their increments, it follows that the aggregate of the four quantities has itself the properties of a vector; these quantities are referred to an instantaneous condition of the body (e.g. at the time = ). This 4-vector may also be expressed in terms of the mass , and the velocity of the body, considered as a material particle. To form this expression, we note first, that