On The Electrodynamics Of Moving Bodies Part:6

§ 5. The Composition of Velocities

In the system k moving along the axis of X of the system K with velocity v, let a point move in accordance with the equations

where w_ξ and w_η denote constants.

Required: the motion of the poinrelatively to the system K. If with the help of the equations of transformation developed in § 3 we introduce the quantities x, y, z, t into the equations of motion of the point, we obtain

Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation. We set

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⁷Not a pendulum-clock, which is physically a system to which the Earth belongs. This case had to be excluded.

^†Editor’s note: This equation was incorrectly given in Einstein’s original paper and the 1923 English translation as a = tan⁻¹ w_y/w_x

a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain

It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get

It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c. For if we set v = c − κ, w = c − λ, κ and λ being positive and less than c, then

It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain

We might also have obtained the formula for V, for the case when v and w have the same direction, by compounding two transformations in accordance with § 3. If in addition to the systems K and k figuring in § 3 we introduce still another system of co-ordinates k' moving parallel to k, its initial point moving on the axis of Ξ^† with the velocity w, we obtain equations between the quantities x, y, z, t and the corresponding quantities of k', which differ from the equations found in § 3 only in that the place of “v” is taken by the quantity

from which we see that such parallel transformations—necessarily—form a group. We have now deduced the requisite laws of the theory of kinematics cor- responding to our two principles, and we proceed to show their application to electrodynamics.

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