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 u_{x} = u in the same direction.
Then if the electric and magnetic vectors are proportional to e^{iA(x - vt)}, we have
[part]/[part]x = iA, [part]/[part]t = -iAv, [part]/[part]y = [part]/[part]z = 0, u_{y} = u_{z} = 0
Then [part]Dy/[part]t = -c[part]H_{z}/[part]x - u[part]D_{y}/[part]z . . . (1·21)
and [part]B_{z}/[part]t = -c[part]E_{y}/[part]x - u[part]B_{z}/[part]x (2·21)
Since D = KE and B = μH, we have
iAv(κEy) = -ciA(H_{z} + uKE_{y}) (1·22)
iAv(μH_{z}) = -ciA(E_{y} + uμH_{z}) (2·22)
or v(K - u)E_{y} = cH_{z} (1·23)
μ(v - u)H_{z} = cE_{y} (2·23)
Multiplying (1·23), by (2·23)
μK(v - u)^2 = c^2
Hence (v - u)^2 = c^2/μk = v_{0}^2
[therefore] v = v_{0} + u,
making Fresnelian convection co-efficient simply unity.
Equations (1·21), and (2·21) may be obtained more simply from physical considerations.
According to Heaviside and Hertz, the real seat of both electric and magnetic polarisation is the moving medium itself. Now at a point which is fixed with respect to the ether, the rate of change of electric polarisation is δD/δt.
