Gravitation and Light 339 



process of continuation, it will prove to be just the same surface 

 as before. Whether it is expressed in terms of p' , q' or of jp, q is 

 intrinsically of no consequence : the coordinates are of no account, 

 it is only the functional forms of/, g, h that are essential. 



This last statement, developed in terms of the criterion of 

 invariance in order to avoid a representation by immersion in a 

 uniform geometrical manifold of dimensions higher than the given 

 four of space and time, appears to cover the general relativity 

 of Einstein. The/, g, h, ... can be named the potentials which deter- 

 mine the space. In the special relativity, before gravitation was 

 absorbed into the metric of extension, all spaces were flat, so 

 /, g, h, ... were constants ; which is all that is left, for that particular 

 case, of these relations of invariance. 



In this flat fourfold, relativity implied merely that a physical 

 system is determined by its own internal relations, so that the 

 position that may be assigned to it in the fourfold is of no account, 

 any more than is the position of a surface or a system of bodies 

 in space. In the later general relativity the manifold must be 

 supposed given descriptively by coordinates, which represent 

 numerical counts arranged to suit the number of dimensions that 

 are involved : it only gains internal form when a metric is imposed 

 upon it. If the Euchdean metric 



§s^ = Sp^ + Sq^+ ... 



is imposed it becomes a Euclidean space everywhere uniform and 

 also flat, in which bodies are mobile without change of form. If 

 a metric varying with position is imposed, the expressions in this 

 manifold of the metric relations of nature will become complicated, 

 and the relations so changed be described as a modified set of laws. 



The original non-metric continuum might be marked for 

 instance by gradations of colour: the colour-scheme of Newton as 

 developed by Young, Helmholtz, and Maxwell, is the standard 

 example of a non-metric threefold extension. 



May we not here have refined down to the unresolvable essence 

 of space, as the mere possibihty of descriptive continuity of three- 

 fold type which is an essential feature in our mental world ? Within 

 this a priori datum of threefold uncharted pure continuity we may 

 construct types of charted spaces almost without limit, by imposing 

 metrics of various types. Any particular space is not however 

 determined by the system of coordinates of reference p, q, ... but by 

 the variable coeSicients f, g, h, ... of the imposed metric expressed 

 as functions of them. But yet it is only under special conditions 

 when it is uniform and flat that finite difl'erences of these co- 

 ordinates can be involved, this being part of the expression of the 

 mobility of solid bodies in the space. It is in this narrower sense, 

 that "the system of coordinates is accidental, that relativity has 



