^{page 183 note *} Proc. Boy. Soc., A, 133 (1931), 93.

^{page 183 note †} The instantaneous three-dimensional space of the observer consists of those world-points in his immediate neighbourhood which he regards as simultaneous. Geometrically, it is a small portion near O of the hypersurface formed by the geodesics through O which are perpendicular at this point to the observer's world-line.

^{page 186 note *} The η's are in fact a particular set of Riemann normal coordinates. Systems of this type have recently been employed by Thomas, T. Y., Proc. Nat. Acad. Sci., 16 (1930), 761.CrossRefGoogle Scholar

^{page 187 note *} Eddington, , Mathematical Theory of Relativity (1924), 163.Google Scholar

^{page 187 note †} See, for example, Eisenhart, , Riemannian Geometry (1926)Google Scholar, ch. iii, (29.3).

^{page 188 note *} Veblen, , Invariants of Quadratic Differential Forms (Camb. Math. Tract No. 24, 1927)Google Scholar, ch. vi.

^{page 188 note †} Ruse, , Proc. London Math. Soc., 32 (1931), 90.Google Scholar

^{page 189 note *} It is necessary to assume that he measures time by a clock in his possession, so that the physical time is identical with his proper-time τ.

^{page 191 note *} x = x, y = y, z = z are then the equations of a geodesic, since the equations of geodesies in this space are all of the form x = au + b, y = a′u + b′, z=a″u + b″, where the a's and b's are constants.

^{page 191 note †} Whittaker, loc. cit., equation (4).

^{page 193 note *} Whittaker, , loc. cit., 96Google Scholar, and Eddington, , Mathematical Theory of Relativity (1924), 161.Google Scholar

^{page 194 note *} Op. cit., 163.