* The line-element ds is considered as an infinitesimal vector, tangent to the curve; the surface-element dσ as an infinitesimal vector normal to the surface. The space-element is denoted by dS, the space-time element by dΣ.

* The indices h, i, j, k, l, m run through the values 1, 2, 3, 4.

† Cf. Cunningham, E., Proc. London Math. Soc. (2), 8 (1910), 77–98CrossRefGoogle Scholar; Bateman, H., ibidem 8 (1910), 223–264, 469–488Google Scholar; 21 (1920), 256–270. Compare also: Bessel-Hagen, E., “Über die Erhaltungssätze der Elektrodynamik,” Math. Ann. 84 (1921), 258–276.CrossRefGoogle Scholar

‡ Schouten, J. A. and Haantjes, J., “Über die konforminvariante Gestalt der Maxwellschen Gleichungen und der elektromagnetischen Impulsenergiegleich-ungen,” Physica, 1 (1934), 869–872CrossRefGoogle Scholar ("Konforme Feldtheorie i").

§ The indices a, b, c, d, e, f, g will run through the values 1, 2, 3 only.

* a _ab is the fundamental tensor of space: a _ab = g _ab. Note that a _ab is negative definite.

† e′_abc is the covariant unity trivector:

e ^abc is the contravariant unity trivector:

Hence e′_abc = − e _abc, where the indices are lowered with the aid of a _ab.

‡ Space coordinates are denoted by ξ^a, space-time coordinates by ξⁱ. For orthogonal coordinates we have

§ We use, with Schouten, square brackets to denote the alternated part of an affinor, e.g.

∥ Compare, however, the equivalent form, mentioned below.

* Hence

* The following identities (formulated for n-vector densities in n-dimensional space) are often used in computations:

where is defined by

In particular, we often use

† The equation I = λE is irrelevant, since it is only an asymptotic condition, valid for stationary currents.

* I hope to return to this question on a later occasion.

† It might seem that by means of η_ijkl metric (though a generalized form of it) is introduced in the theory again. This, however, is not the case: in the ordinary theory matter is characterized by μ^ab, ε^ab and metric by a ^ab; they all occur in the formulae (1), (14). We only use η_ijkl which characterizes matter, but has nothing to do with ordinary metric, except in special cases. It defines a kind of conformal metric for surface-elements dσ ^ij, determined by the scalar density of weight − 1

If and only if η_ijkl degenerates according to (27) this conformal metric can be extended to a conformal metric for line-elements dξ ⁱ, determined by the scalar density of weight

Nevertheless in permeable or dielectric matter these metrics are still independent of the ordinary conformal metric, which is merely an extrapolation of the conformal metric of free ether into matter, so that we do not return to the dualistic standpoint of ordinary electrodynamics.