Page:The principle of relativity (1920).djvu/172

 (x_{1} x_{2} x_{3} x_{4}) such that to any and every point-event corresponds a system of values of (x_{1} x_{2} x_{3} x_{4}). Two coincident point-events correspond to the same value of the variables (x_{1} x_{2} x_{3} x_{4}); i.e., the coincidence is characterised by the equality of the co-ordinates. If we now introduce any four functions (x´_{1} x´_{2} x´_{3} x´_{4}) as co-*ordinates, so that there is an unique correspondence between them, the equality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence of two material points. As the purpose of all physical laws is to allow us to remember such coincidences, there is a priori no reason present, to prefer a certain co-ordinate system to another; i.e., we get the condition of general covariance. § 4. Relation of four co-ordinates to spatial and time-like measurements. Analytical expression for the Gravitation-field.

I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible, with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonable according to the light of experience. In this sense, we shall now introduce the following supposition; that for an infinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen.

The nature of acceleration of an infinitely small (positional) co-ordinate system is hereby to be so chosen, that the gravitational field does not appear; this is possible for an infinitely small region. X_{1}, X_{2}, X_{3} are the spatial