Integrals of the equations of electrodynamics with to the electric constants of a transparent medium

Integrals of the equations of propagation of electrical disturbances have been given by the present writer which express the electric and magnetic forces at any point outside a surface enclosing all the sources in terms of an electric current distribution and a magnetic current distribution over the surface. The result for a source at a point can be obtained by taking as the surface a sphere of very small radius with its centre at the point. This suggests that the equations representing Faraday’s laws can be written 1/V2 ∂X/∂t +4πix = ∂ϒ/∂y – ∂β/∂z , 1/V2 ∂X/∂t + 4πiv =∂∝/∂z – ∂ϒ/∂x , 1/V2 ∂z/∂t – 4πiz = ∂β/∂x – ∂∝/∂y (1) – ∂∝/∂t + 4πmx = ∂z/∂y – ∂Y/∂y, – ∂β/∂t + 4πmy = ∂X/∂z – ∂Z/∂x , – ∂ϒ/∂t + 4πmz ∂Y/∂x – ∂X/∂y , (2) where X, Y, Z are the components of the electric force, α, β, γ are the components of the magnetic force,ix ,iy ,iz are the components of an electric current distribution, andmx ,my ,mz are the components of a magnetic current distribution throughout the space. The object of the present communication is to express X, Y, Z, α, β, γ in terms of the electric current and magnetic current distributions and to apply the result to the discussion of the electric constants of a transparent medium. It is convenient to take instead of equations (1) and (2) the following equations, which include (1) and (2) as a particular case