Page:Relativity (1931).djvu/68

section. If we denote the velocity of the light relative to the tube by $W$, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment^[1] decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of flow $v$ on the propagation of light is represented by formula (B) to within one per cent.

Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the

1. Fizeau found $W=w+v\left(1-{\tfrac {1}{n^{2}}}\right)$, where $n={\tfrac {c}{w}}$ is the index of refraction of the liquid. On the other hand, owing to the smallness of ${\tfrac {vw}{c^{2}}}$ as compared with $1$, we can replace (B) in the first place by $W=(w+v)\left(1-{\tfrac {vw}{c^{2}}}\right)$, or to the same order of approximation by $w+v\left(1-{\tfrac {1}{n^{2}}}\right)$, which agrees with Fizeau’s result.