Subsequently we saw that accelerated frames moving in interstellar space would be represented by curvilinear or Gaussian mesh-systems. In these systems the values of the ’s of space-time were no longer constant, but varied from place to place. This variation in the values of the ’s connoted the variation in value of the potentials; and as a result the mathematical expression of a force (given as it was by relations between these variations) no longer vanished. These curvilinear mesh-systems represented, therefore, systems in which fields of force would be present. This again was in perfect accord with what we knew of accelerated or non-Galilean frames. Further, as referred to these curvilinear mesh-systems, the geodesics of space-time would appear to be curved. The physical significance of this fact would be to imply that the motion of free bodies and light waves would appear to be accelerated when viewed from accelerated frames. Once more the complete accord with experiment is apparent. Now let us proceed to the natural generalisation of these results. So far, we have assumed space-time to be flat. Indeed, we could not escape this assumption if we were to admit that Galilean frames, free from all fields of force, could exist in space far from matter, and that as observed from these frames the geodesics of space-time would be Euclidean straight lines. Only thus would Newton’s law of the motion of free bodies in interstellar space be verified. But suppose now that, for some reason or other, space-time were to develop non-Euclideanism or curvature, and so depart from its flatness. How would such a condition manifest itself physically? In the first place, it would be impossible to split up this curved space-time into the straight directions which define a Cartesian mesh-system. Physically speaking, this would imply that in those regions where space-time was curved no Galilean systems could exist. We might, of course, conceive of mesh-systems less curved than others; but even the straightest of these systems would at best be only approximately Cartesian within a restricted region. As we considered parts of the mesh-system farther and farther removed from this special region, the meshes would become more and more widely spaced or more and more closely gathered together.