Page:Relativity (1931).djvu/66

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

$x'=wt'$

we must then express $x'$ and $t'$ in terms of $x$and $t$, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation

$W={\frac {v+w}{1+{\frac {vw}{c^{2}}}}}$

(B),

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we are enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since