Ritz proposed a new equation, derived from the principles of the ballistic theory of electromagnetic waves, a theory competing with the special theory of relativity. The equation relates the force between two charged particles with a radial separation r relative velocity v and relative acceleration a, where k is an undetermined parameter from the general form of Ampere's force law as proposed by Maxwell. The equation obeys Newton's third law and forms the basis of Ritz's electrodynamics.
On the assumption of an emission theory, the force acting between two moving charges should depend on the density of the messenger particles emitted by the charges (), the radial distance between the charges (ρ), the velocity of the emission relative to the receiver, ( and for the x and r components, respectively), and the acceleration of the particles relative to each other (). This gives us an equation of the form:
where the coefficients , and are independent of the coordinate system and are functions of and . The stationary coordinates of the observer relate to the moving frame of the charge as follows
Developing the terms in the force equation, we find that the density of particles is given by
The tangent plane of the shell of emitted particles in the stationary coordinate is given by the Jacobian of the transformation from to :
We can also develop expressions for the retarded radius and velocity using Taylor series expansions
With these substitutions, we find that the force equation is now
Next we develop the series representations of the coefficients
With these substitutions, the force equation becomes
Since the equation must reduce to the Coulomb force law when the relative velocities are zero, we immediately know that . Furthermore, to obtain the correct expression for electromagnetic mass, we may deduce that or .
To determine the other coefficients, we consider the force on a linear circuit using Ritz's expression, and compare the terms with the general form of Ampere's law. The second derivative of Ritz's equation is
Diagram of elements of linear circuits
Consider the diagram on the right, and note that ,
Plugging these expressions into Ritz's equation, we obtain the following
From this we obtain the full expression of Ritz's electrodynamic equation with one unknown
In a footnote at the end of Ritz's section on Gravitation (English translation) the editor says, "Ritz used k = 6.4 to reconcile his formula (to calculate the angle of advancement of perihelion of planets per century) with the observed anomaly for Mercury (41") however recent data give 43.1", which leads to k = 7. Substituting this result into Ritz's formula yields exactly the general relativity formula." Using this same integer value for k in Ritz's electrodynamic equation we get:
^O'Rahilly, Alfred (1938). Electromagnetics; a discussion of fundamentals. Longmans, Green and Co. pp. 503–509. OCLC3156160. Reprinted as O'Rahilly, Alfred (1965). Electromagnetic Theory. Dover Books. pp. 503–509.