where u is the vector velocity of the moving body and [Ru] the vector product of R and u.
Hence equations (1) and (2) become
c curl H = ∂D/∂t + u div D + curl Vect. [Du] + ρν (1·1)
and
-c curl E = ∂B/∂t + u div B + curl Vect. [Bu] (2·1)
which gives finally, for ρ = 0 and div B = 0,
∂D/∂t + u div D = c curl (H - 1/c Vect. [Du]) (1·2)
∂B/∂t = -c curl (E - 1/c Vect. [uB]) (2·2)
Let us consider a beam travelling along the x-axis, with apparent velocity v (i.e., velocity with respect to the fixed ether) in medium moving with velocity ux = u in the same direction.
Then if the electric and magnetic vectors are proportional to eiA(x - vt), we have