{ -1, 0, 0, 0 (4) { 0 -1 0 0 { 0 0 -1 0 { 0 0 0 +1 We would afterwards see that the choice of such a system of co-ordinates for a finite region is in general not possible. From the considerations in § 2 and § 3 it is clear, that from the physical stand-point the quantities _g__{στ} are to be looked upon as magnitudes which describe the gravitation-field with reference to the chosen system of axes. We assume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for some particular choice of co-ordinates. The _g__{στ}’s then have the values given in (4). A free material point moves with reference to such a system uniformly in a straight-line. If we now introduce, by any substitution, the space-time co-ordinates _x₁_..._x₄_ then in the new system _g__{μν}’s are no longer constants, but functions of space and time. At the same time, the motion of a free point-mass in the new co-ordinates, will appear as curvilinear, and not uniform, in which the law of motion, will be _independent of the nature of the moving mass-points_. We can thus signify this motion as one under the influence of a gravitation field. We see that the appearance of a gravitation-field is connected with space-time variability of _g__{στ}’s. In the general case, we can not by any suitable choice of axes, make special relativity theory valid throughout any finite region. We thus deduce the conception that _g__{στ}’s describe the gravitational field. According to the general relativity theory, gravitation thus plays an exceptional rôle as distinguished from the others, specially the electromagnetic forces, in as much as the 10 functions _g__{στ} representing gravitation, define immediately the metrical properties of the four-dimensional region. B Mathematical Auxiliaries for Establishing the General Covariant Equations. We have seen before that the general relativity-postulate leads to the condition that the system of equations for Physics, must be co-variants for any possible substitution of co-ordinates _x₁_, ... _x₄_; we have now to see how such general co-variant equations can be obtained. We shall now turn our attention to these purely mathematical propositions. It will be shown that in the solution, the invariant _ds_, given in equation (3) plays a fundamental rôle, which we, following Gauss’s Theory of Surfaces, style as the line-element.