It should be noticed that v and w enter into the expression for velocity symmetrically. If w has the direction of the ξ-axis of the moving system,
From this equation, we see that by combining two velocities, each of which is smaller than c, we obtain a velocity which is always smaller than c. If we put v = c - χ, and w = c - λ, where χ and λ are each smaller than c,[[8]]
It is also clear that the velocity of light c cannot be altered by adding to it a velocity smaller than c. For this case,
We have obtained the formula for U for the case when v and w have the same direction; it can also be obtained by combining two transformations according to section [§ 3]. If in addition to the systems K, and k, we introduce the system k´, of which the initial point moves parallel to the ξ-axis with velocity w, then between the magnitudes, x, y, z, t and the corresponding magnitudes of k´, we obtain a system of equations, which differ from the equations in [§ 3], only in the respect that in place of v, we shall have to write,
We see that such a parallel transformation forms a group.