corresponding to the material point, we thus have a (uni-dimensional) line in the four-dimensional continuum.  In the same way, any such lines in our continuum correspond to many points in motion.  The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters.  In our mathematical treatment, such an encounter is expressed in the fact that the two lines which represent the motions of the points in question have a particular system of co-ordinate values, x[1], x[2], x[3], x[4], in common.  After mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements.

When we were describing the motion of a material point relative to a body of reference, we stated nothing more than the encounters of this point with particular points of the reference-body.  We can also determine the corresponding values of the time by the observation of encounters of the body with clocks, in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials.  It is just the same in the case of space-measurements by means of measuring-rods, as a litttle consideration will show.

The following statements hold generally :  Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates x[1], x[2], x[3], x[4].  Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.

EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY

We are now in a position to replace the pro. visional formulation of the general principle of relativity given in Section 18 by an exact formulation.  The form there used, “All bodies of reference K, K1, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion,” cannot be maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description.  The Gauss co-ordinate system has to take the place of the body of reference.  The following statement corresponds to the fundamental idea of the general principle of relativity:  “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”