The Einstein addition law can also be obtained by remembering that the second velocity is...

The Einstein addition law can also be obtained by remembering that the second velocity is related directly to the space components of a four-velocity, which may then be transformed back to the initial system by a Lorentz transformation. If the second system is moving with a speed v′ relative to the first in the direction of their z axes, while a third system is moving relative to the second with an arbitrarily oriented velocity v″, show by this procedure that the magnitude of the velocity v between the first and third system is given by

and that the components of v are

Here βx″ = vx″/c, and so forth.