To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint. PART II. ELECTRO-MAGNETIC PHENOMENA. § 7. Fundamental Equations for bodies at rest. After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limiting case ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us—when proper fundamental data are given—to obtain the following quantities at every place and time, and therefore at every space-time point as functions of (_x_, _y_, _z_, _t_):—the vector of the electric force E, the magnetic induction M, the electrical induction _e_, the magnetic force _m_, the electrical space-density ρ, the electric current s (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector _v_, the velocity of matter. The relations in question can be divided into two classes. Firstly—those equations, which,—when _v_, the velocity of matter is given as a function of (_x_, _y_, _z_, _t_),—lead us to a knowledge of other magnitude as functions of _x_, _y_, _z_, _t_—I shall call this first class of equations the fundamental equations— Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector _u_ as functions of (_x_, _y_, _z_, _t_). For the case of bodies at rest, _i.e._ when _u_ (_x_, _y_, _z_, _t_) = 0 the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are;— (1) The Differential Equations:—which contain no constant referring to matter:— (_i_) Curl _m_ - δ_e_/δ_t_ = C, (_ii_) div _e_ = lρ. (_iii_) Curl E + δM/δ_t_ = 0, (_iv_) Div M = 0. (2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves _i.e._ for isotopic bodies;—they are comprised in the equations (V) _e_ = ε E, M = μ_m_, C = σE. where ε = dielectric constant, μ = magnetic permeability, σ = the conductivity of matter, all given as function of _x_, _y_, _z_, _t_; _s_ is here the conduction current. By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work, _x₁_ = _x_, _x₂_ = _y_, _x₃_ = _z_, _x₄_ = _it_,