in which the indices and run from 1 to 3 only. The will be such functions of , , as correspond to a three-dimensional continuum of constant positive curvature. We must now investigate whether such an assumption can satisfy the field equations of gravitation. In order to be able to investigate this, we must first find what differential conditions the three-dimensional manifold of constant curvature satisfies. A spherical manifold of three dimensions, [Pg 110] embedded in a Euclidean continuum of four dimensions,[19] is given by the equations By eliminating , we get [19]The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice. As far as terms of the third and higher degrees in the , we can put, in the neighbourhood of the origin of co-ordinates, Inside the brackets are the of the manifold in the neighbourhood of the origin. Since the first derivatives of the , and therefore also the , vanish at the origin, the calculation of the for this manifold, by (88), is very simple at the origin. We have Since the relation is universally co-variant, and since all points of the manifold are geometrically equivalent, this relation holds for every system of co-ordinates, and everywhere in the manifold. In order to avoid confusion with [Pg 111] the four-dimensional continuum, we shall, in the following, designate quantities that refer to the three-dimensional continuum by Greek letters, and put We now proceed to apply the field equations (96) to our special case. From (119) we get for the four-dimensional manifold, For the right-hand side of (96) we have to consider the energy tensor for matter distributed like a cloud of dust. According to what has gone before we must therefore put specialized for the case of rest. But in addition, we shall add a pressure term that may be physically established as follows. Matter consists of electrically charged particles. On the basis of Maxwell's theory these cannot be conceived of as electromagnetic fields free from singularities. In order to be consistent with the facts, it is necessary to introduce energy terms, not contained in Maxwell's theory, so that the single electric particles may hold together in spite of the mutual repulsions between their elements, charged with electricity of one sign. For the sake of consistency with this fact, Poincaré has assumed a pressure [Pg 112] to exist inside these particles which balances the electrostatic repulsion. It cannot, however, be asserted that this pressure vanishes outside the particles. We shall be consistent with this circumstance if, in our phenomenological presentation, we add a pressure term. This must not, however, be confused with a hydrodynamical pressure, as it serves only for the energetic presentation of the dynamical relations inside matter. In this sense we put In our special case we have, therefore, to put