What is the simplest derivation of the transformation rules for Maxwell's equations in special relativity? I'm working through Einstein's original 1905 paper(available here), and I'm having trouble with the section on the transformation of Maxwell's equations from rest to moving frame. The paper proceeds as follows: I do not understand just where this second set of equations comes from. The only derivations of this transformation that I have been able to find involve either potentials, four-vectors, or both. However since four-vectors had not been invented in 1905, and because the statement of the transformation is so blunt, it seems that Einstein is using or appealing to a simpler method for finding the transformation. So my question is: What is the simplest derivation of the transformation rules for Maxwell's equations in special relativity? Is Einstein appealing to a pre-existing Lorentz method here, or is there a trick to accomplish all of this quickly? Or does the text here simply belie the true work involved in the derivation?