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 x_{1}, . . . x_{4}; 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.
The fundamental idea of the general co-variant theory is this:—With reference to any co-ordinate system, let certain things (tensors) be defined by a number of functions of co-ordinates which are called the components of
