denotes the sun's mass, centrally symmetrically placed about the origin of co-ordinates; the solution (109) is valid only outside of this mass, where all the vanish. If the motion of the planet takes place in the plane then we must replace (109a) by [Pg 101] The calculation of the planetary motion depends upon equation (90). From the first of equations (110b) and (90) we get, for the indices 1, 2, 3, or, if we integrate, and express the result in polar co-ordinates, From (90), for = 4, we get From this, after multiplication by and integration, we have In (109c), (111) and (112) we have three equations between the four variables , , and , from which the motion of the planet may be calculated in the same way as in classical mechanics. The most important result we get from this is a secular rotation of the elliptic orbit of the planet in the same sense as the revolution of the planet, amounting in radians per revolution to [Pg 102] where This expression furnishes the explanation of the motion of the perihelion of the planet Mercury, which has been known for a hundred years (since Leverrier), and for which theoretical astronomy has hitherto been unable satisfactorily to account. There is no difficulty in expressing Maxwell's theory of the electromagnetic field in terms of the general theory of relativity; this is done by application of the tensor formation (81), (82) and (77). Let be a tensor of the first rank, to be denoted as an electromagnetic 4-potential; then an electromagnetic field tensor may be defined by the relations, The second of Maxwell's systems of equations is then defined by the tensor equation, resulting from this, and the first of Maxwell's systems of equations is defined by the tensor-density relation [Pg 103] in which If we introduce the energy tensor of the electromagnetic field into the right-hand side of (96), we obtain (115), for the special case = 0, as a consequence of (96) by taking the divergence. This inclusion of the theory of electricity in the scheme of the general theory of relativity has been considered arbitrary and unsatisfactory by many theoreticians. Nor can we in this way conceive of the equilibrium of the electricity which constitutes the elementary electrically charged particles. A theory in which the gravitational field and the electromagnetic field enter as an essential entity would be much preferable. H. Weyl, and recently Th. Kaluza, have discovered some ingenious theorems along this direction; but concerning them, I am convinced that they do not bring us nearer to the true solution of the fundamental problem. I shall not go into this further, but shall give a brief discussion of the so-called cosmological problem, for without this, the considerations regarding the general theory of relativity would, in a certain sense, remain unsatisfactory.