There can be no doubt that of the existence of aberration as a numerical fact, but whether Bradley was rightly guided to its cause by the the flag on its mast-head of the boat is less certain.
It is, of course, one of the triumphs of the Special Theory of Relativity that it gives a logical and satisfactory explanation of this and similar experiments, notably Fizeau's water-tube experiment […]
There is a concise account of the phenomenon from the Relativity standpoint on p. 21 of Eddington's Mathematical Theory of Relativity. In Relativity theory, Airy's experiment is trivial, as the observer is at rest relative to the telescope.