By J. Aharoni

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W) + p••••w >

Cf. 27) and GRASSMANN 1844, 1, p. 206; 1862, 1, p. 56; ROTHE 1912, 1, p. 1039. 3 ) The results obtained under A, C and D of this section give a possibility of classification for cocontravariant and symmetric tensors of valence 2 and for bivectors. The problern of the classification of quantities with a valence >2 is only solved in very special cases. For trivectors the classification was given for n = 6 by REICHEL 1907, 1, for n = 7 by SCHOUTEN 1931, 1 and for n = 8 by GUREWITSCH 1935, 1. Cf. W.

Every tensor P";. of rank r can of course be written in the form px?. 25) = u"v;. 1 1 + ··· + u"v;.. r r The vectors u" span the u-domain and the vectors v;. -domain. The same holds mutatis mutandis for the covariant case. 26) S! p[><, [A, ... p>~~< JA, I • • • p>~~~~< [Ä p'>~~~~ r but not for s = r. 26) is invariant for the operation of interchanging the set of indices :><:1 , ... ~~