Friedrich Kottler

“ The supplement in an Euclidean is of course nothing other than the duality of polar correlation in the infinitely distant with respect to the absolute of Euclidean metric, therefore the name dual system is used.
As the reciprocal system [8] , on the other hand, we denote the following covariant or contravariant system, which emerges from the contravariant or covariant system of same order be means of the coefficients of :
with the solution
,
where the are not forming a special system this time. ”

“ Difference of the components of a vector determined by the differential invariant theory compared to the ordinary representation. The latter one operates by the method of vectorial splitting with respect to the directions of the -gon, which form in any spacepoint the passing parameter lines. So, let be Cartesian orthogonal coordinates in and be any generalized coordinates, then by (2) it is given:
and in consequence of (3) :
So if one has a covariant system of -th order in Cartesian coordinates , then it is given ”

“ The normal-plane , which definitely belongs to the framing- by virtue of the Euclidean metric, will be oriented correctly, if one predefines a direction within it in an arbitrary way, and in this way determines the necessary second direction : One intersects the normal plane with the curvilinear , by which an intersection curve emerges which forms with an even number of intersections ”

“ The polar vectors of are contravariants of first order, the axial vectors are special contravariants of second order, from which the so-called supplement (a covariant system of first order) has to be derived.
Supplement. We define the following covariant system -th order as the supplement of a contravariant system of -th order:
Let be a positive permutation of , then for the supplement of the system (which is assumed to be special) , we have
with the remark that in the case of motions changing the direction, i.e. ”

“ Wright, Invariants of quadratic differential forms. Cambridge Tracts in Mathematics etc. No 9, chapter II.↑ Covariant or contravariant systems of -th order will be marked by lowered or raised indices. The themselves are not contravariants.↑ Christoffel, Crelle's Journal, 70, p. 46 (1869) ”