† Math. Annalen, 36 (1890), 189.Google Scholar

‡ Atti Acc. Torino, 40 (1905), 3–17.Google Scholar Levi's proof depends upon Cartan's, E. classification of semi-simple groups (Thèse, Paris, 1894, 2nd ed. 1933). In his thesis Cartan proved the theorem in certain special cases (Chap. VI and p. 128, 2nd ed.).Google Scholar

§ See Cartan, , loc. cit. Chap. VI, theorem I.Google Scholar

║ Cf. Weyl, H., Math. Zeit. 23 (1925), 275.CrossRefGoogle Scholar The three parts of this paper, continued in Math. Zeit. 24 (1926), 328–95,CrossRefGoogle Scholar wi11 be referred to as Weyl I, II and III. For a general account of the structure of an infinitesimal group we Cartan, loc. cit., or Eisenhart, L. P., Continuous groups of transformations, Princeton (1933), Chap. IV.Google Scholar

¶ Roman indices take the values 1, …, n and Greek indices, unless otherwise stated, the values, I, …, r.

† Indices will be raised and lowered in the ordinary way by means of the tensor gαβ, whose determinant does not vanish since the group g is semi-simple—Cartan, loc. cit. Chap. IV, theorem 1. In the absence of dots it is to be understood that indices are raised from the first lower to the last upper index and vice versa.

† Eisenhart, , loc. cit. p. 162.Google Scholar

‡ Since implies υ_σ = 0 the equations (1.2) have at most one set of solutions.

§ Cartan, , loc. cit. (2nd ed.), p. 113.Google Scholar

║ Notice that , regarded as a set of r vectors in a space of ½–r (r−1) dimensions, constitute a complete set of (independent) solutions of (1·5b), regarded as linear equations in the variables A _λμ.

† Weyl, II, p. 375.Google Scholar

† Weyl, III, p. 381 and II, p. 289.Google Scholar See also Cartan, , La théorie des groupes finis et continus et l'analysas situs, Paris (1930), pp. 37–41.Google Scholar

‡ By a real subspace is meant one which contains the vector if it contains x It is not difficult to show that such a subspace is spanned by a set of vectors with real co-ordinates.

† Cartan, Thèse, Chap.IV, theorem 4.

‡ In other words, E/E₁ is obtained from E by identifying any two vectors whose difference lies in E₁.

† Weyl, III, pp. 360–1 and theorem 2, p. 364.Google Scholar The argument on p. 278 of Weyl I shows that Λ+α is a weight for at least one root a if Λ is a given weight.

† An equi-affine space of paths is one in which where ρ is a scalar density which may be taken to define volume. A necessary and sufficient for this to be the case is B_jk = B_kj where (see Eisenhart, L. P., Non-Riemannian geometry, New York (1929), p. 9).Google Scholar

‡ Ibid. p. 126.

† Cf. Hopf, H. and Rinow, W., Comment. Math. Heiv. 3 (1931), 209–25.CrossRefGoogle Scholar

‡ It is now to be understood that all the entities to which we refer are analytic and defined all over V_n. In particular each individual vector ξλ is defined at each point, and the operators (h₁,…, h_r) generate a finite group in V_n.

§ Actually h_λf is a normal function whether f is normal or not.

║ See a forthcoming paper by Hodge, W. V. D., Journal London Math. Soc. 11 (1936).Google Scholar