In the 1860's Maxwell formulated what are now called Maxwell's equation, and he found that they lead to a remarkable conclusion: the existence of electromagnetic waves that propagate at a speed $c$, which turns out to be the speed of light, implying that light is an electromagnetic wave. Now the fact that Maxwell's equations predict speed of light is $c$ suggested to Maxwell and others that Maxwell's equations are not actually true in all frames of reference. Instead, they thought, Maxwell's equations only exactly true in one frame, the rest frame of the aether, and in all other frames they would have to be replaced by other equations, equations that were invariant under Galilean transformations in order to conform to the principle of relativity. These other equations implied that the speed of light in other frames was actually $c+v$ or $c-v$, where $v$ is the speed of the aether. But then the Michelson-Morley experiment, which was intended to find the speed $v$ of the aether, ended up showing that the speed of light was $c$ in all frames, apparently contradicting the principle of relativity. But Einstein showed that this doesn't contradict the principle of relativity at all, it's just that you need to rethink your notions of space and time.
But my question is, what are the equations that people thought were true in frames other than the aether frame? To put it another way, what are the equations you obtain if you apply a Galilean transformation to Maxwell's equations? (As opposed to a Lorentz transformation which leaves Maxwell's equations unchanged.)
I've actually seen the equations obtained before. They were formulated by some 19th century physicist, maybe Hertz or Heaviside, and they involve adding velocity-dependent terms to the Ampere-Maxwell law and Faraday's law. (Dependent on the velocity of aether, that is.) But I don't remember the details.