We may call this vector R the moving force of the material point.
If instead of integrating over the normal section, we integrate the equations over that cross section of the filament which is normal to the t axis, and passes through (x, y, z, t), then [See (4)] the equations (22) are obtained, but
are now multiplied by dτ/dt; in particular, the last equation comes out in the form,
m d/dt (dt/dτ) = wx Rx dτ/dt + wy Ry dτ/dt + wz Rz dτ/dt.
The right side is to be looked upon as the amount of work done per unit of time at the material point. In this equation, we obtain the energy-law for the motion of the material point and the expression
m (dt/dτ - 1) = m [1/√(1 - w²) - 1]
= m (½ |w₁² + 3/8 |w₁⁴ + )
may be called the kinetic energy of the material point.
Since dt is always greater than dτ we may call the quotient (dt - dτ)/dτ as the “Gain” (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed;—The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.
The set of four equations (22) again shows the symmetry in (x, y, z, t), which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. On the ground of this demand for symmetry, the triplet consisting of the first three equations are to be constructed after the model of the fourth; remembering this circumstance, we are justified in saying,—