D O C . 3 1 3 O N C U R R E N T S T AT E O F F I E L D T H E O R Y 3 0 3 Now, we wish to demonstrate the existence of distant parallelism. At one point P 0 , we can freely choose the orientation of the local n-Bein orthogonal frame. It is then uniquely determined at all the other points on the continuum by the choice that all the corresponding axes of all the local n-Bein frames should be parallel to each other. Parallel vectors then simply have equal local components. For the parallel displacement of a vector A from one point P to another infinitesimally closely neighboring point P ', we thus find the formula , (8) or, from (5), (8), and (6), [14] Then, now setting , [15] (9) we find that the law of parallel displacement is given by . (10) These quantities Δ are in a certain sense analogous to the Christoffel symbols used in Riemannian geometry, in that they are also the coefficients of a paral- lel displacement law. But precisely for these quantities, the contradictory nature of the two structures can be seen. The Riemannian are symmetric in their lower in- dices, but the displacement that they express is not integrable. The Δ are in contrast not symmetric, but the displacement that they express is integrable. The quantities Δ do not have tensor character, but the antisymmetric quantities that can be formed from them, , [16] (11) do have tensor character. By contraction of this tensor, we obtain the vector , which plays the role of the electromagnetic potential in the application of the theory.[17] The exis- tence of a tensor implies that invariants also exist that are formed from the h and their first derivatives. The simplest laws that apply to such a continuum can be found in the following manner: One takes a linear combination (12) of the three invariants[18] A a 0 = A h a A a h a x ---------- A a x h a x --------- - h a A x = = = = h a h a x --------- -Ax h----------Ax a h a x – = h a h a h a ah x -------- - = = A A x – = = – = J A J 1 B J 2 C J 3 + + = [p. 132]