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In Section VI we derived the theorem of the addition of

velocities in one direction in the form which also results from

the hypotheses of classical mechanics. This theorem can also be

deduced readily from the Galilei transformation (Section XI). In

place of the man walking inside the carriage, we introduce a

point moving relatively to the co-ordinate system K′ in

accordance with the equation

x′ = wt′

By means of the first and fourth equations of the Galilei

transformation we can express x′ and t′ in terms of x and t, and

we then obtain

x = (v + w)t

This equation expresses nothing else than the law of motion of

the point with reference to the system K (of the man with

reference to the embankment). We denote this velocity by the

symbol W, and we then obtain, as in Section VI,

W = v + w . . . . . . . (A).

But we can carry out this consideration just as well on the basis

of the theory of relativity. In the equation

x′ = wt′