Pete,
Abraham Marcus 1968 'Electricity for Technicians' and second edition 1975 by Rebecca B. Marcus and then co-author Charles M. Thomson published by Prentice-Hall, Int'l et al; is the very best treatment of magnetism and electricity that I ever read. Brief concise history of discovery and step by natural logical step in the development of theory and application of Electro-Motive Force. The very stuff of Faraday.
Marcus doesn't touch on Maxwell, but all the essentials to understand what Maxwell actually did in his integrating over partial differentials of field intensity, is there. And what gets lost in all the curls and div.s as Maxwell is usually presented, is that while Maxwell and the majority of physics saw 'fields' as disturbances in a media, Faraday saw them as projection in a real material sense of the substance of which they were associated, not as something separate.
I didn't grasp Maxwell until I chanced upon a graphical representation which shows the 'right hand rule' plotted on a Cartesian orthogonal. The direction of motive force is always at right angle between electrical characteristic influence and that of magnetic, but which undergoes a phase transformation relative to velocity of a conductor crossing a field. The rise and fall of respective field intensity and reversing polarity is plotted as a sinusoidal curve intersecting a timeline z-axis. It simplifies to two symmetric inverted 'U' curves, plotted one for electric on (say) x-plane and magnetic on the y-plane. Extrapolated to rest, static state, the magnetic intensity is a 'c'
proportion greater in motive force than the associated magnetic intensity, though both have equal areas under the curve. BUT, the least value of intensity of one field curve will intersect the z-axis at a point incident to the greatest value of intensity of the other field curve. As the comparatively slow speeds of laboratory electromechanical apparatus increases in velocity, that 90 degree displacement of phase decreases. At light velocity the orientation observed in electromotive experiments was also observed in Hertzian waves, but the phase displacement was gone and the slopes of the curves are in sync. What is distinct to any electromagnetic wave of any wavelength/frequency is that the measured greatest intensity of both electric and magnetic motive force is equal. At light velocity electrical field strength has diminished from a 'c' proportion greater than magnetic, to equivalent. AND, the true measured intensity of magnetic motive force is equal to that of a static magnetic field with an associated 'c' proportion greater electrical static field intensity.
This permutates algebraically to a rate of change with velocity as an exponential function. The basis points are there in the increments of lengths in the 90* displacement. The half length on the z-axis to the full length, the half length being one of three equal increments, are the first two terms in an exponential progression; 1/2 + 1/3 + 1/n.
Lorentz does not have that degree of freedom, it is only of time and space. The form of the transform as ; sqrt 1 - (v^2/c^2) reduces to a harmonic series of additive properties; 1/2 + 1/4 +1/2n.
The equivalence function is the same for both, however; 1c (sub) t + 1c (sub) s, = 1c^2.
So if you look at the end-on cosine graphic of your helix you see a rotation of the x and y planes away from the z-axis, but with one co-incidental antipodal point of least intensity, lying on the z-axis incident to a point in time, while the other antipodal point transits through space across time tracking you helix.
Hope this helps find a definitive Theroem of your DFM. jrc
P.S. So yes, there is mathematical form in Maxwell to theorize the proposition of energy density varying as a direct inverse proportion of velocity following an exponential rate of change.
Which is a good point of departure from the prior (lengthy) thread. jrc