2. It must be linear and homogeneous in these second differential coefficients. 3. Its divergence must vanish identically. The first two of these conditions are naturally taken from Poisson's equation. Since it may be proved mathematically that all such differential tensors can be formed algebraically (i.e. without differentiation) from Riemann's tensor, our tensor must be of the form in which and are defined by (88) and (89) respectively. Further, it may be proved that the third condition requires a to have the value . For the law of the gravitational field we therefore get the equation Equation (95) is a consequence of this equation. denotes a constant, which is connected with the Newtonian gravitation constant. In the following I shall indicate the features of the theory which are interesting from the point of view of physics, using as little as possible of the rather involved mathematical method. It must first be shown that the divergence of the left-hand side actually vanishes. The energy principle for matter may be expressed, by (83), [Pg 89] in which The analogous operation, applied to the left-hand side of (96), will lead to an identity. In the region surrounding each world-point there are systems of co-ordinates for which, choosing the -co-ordinate imaginary, at the given point, and for which the first derivatives of the and the vanish. We shall verify the vanishing of the divergence of the left-hand side at this point. At this point the components vanish, so that we have to prove the vanishing only of Introducing (88) and (70) into this expression, we see that the only terms that remain are those in which third derivatives of the enter. Since the are to be replaced by , we obtain, finally, only a few terms which may easily be seen to cancel each other. Since the quantity that we have formed has a tensor character, its vanishing is proved for every other system of co-ordinates also, and naturally for every other four-dimensional point. The energy principle of matter (97) is thus a mathematical consequence of the field equations (96). In order to learn whether the equations (96) are consistent with experience, we must, above all else, find out whether they [Pg 90] lead to the Newtonian theory as a first approximation. For this purpose we must introduce various approximations into these equations. We already know that Euclidean geometry and the law of the constancy of the velocity of light are valid, to a certain approximation, in regions of a great extent, as in the planetary system. If, as in the special theory of relativity, we take the fourth co-ordinate imaginary, this means that we must put