1. Cartan [1, 2].
2. Shorter methods for the determination of Cartan’s coefficients have been given at various times, in particular by Varga [2], Laugwitz [3] and Sulanke [1]. However, since these authors use special devices such as the osculating Riemannian space (see § 4) or the general geometry of paths (in the sense of Douglas [1]), their methods might not exhibit the basic ideas of Cartan as clearly as the original method of the latter. It might, therefore, be more advantageous to follow the historical approach and to postpone the discussion of these more recent constructions. Also, it should be pointed out that Schouten and Haantjes [1] gave a more general theory for the determination of the connection coefficients of spaces in which the fundamental metric function depends on со- and contravariant vector densities. In this connection the reader should also consult Schouten and Hlavatý [1].
3. In point of fact, Cartan gives a further postulate in addition to conditions (a) to (d) ([1], p. 10); however, this postulate is superfluous since it may be derived from the others (see Cartan [3]). It merely involves the relation between perpendicularity and transversality with respect to the element of support.
4. It should be stressed, however, that the notion of length as defined in Chapter I is in general not identical to that defined by (1.1’), the identity holding only if the direction of the vector X
i
coincides with its own element of support.
5. Cartan [1], p. 15.