Relativity of -Field and Matter. 803 



must be extraneous to the electron and involved in its laws 

 o£ formation. It can only belong to (i. e. be determined 

 ■by) the field-entity. There is thus a locus belonging to the 

 field-entity which is similar to and in a constant ratio to 

 the surface of the electron, which in turn is similar to and 

 in a constant ratio to the metre-sphere. Hence the locus 

 in the field-entity is a sphere of constant size, according 

 to our standard of measurement. The field accordingly 

 has at least one property which is everywhere homogeneous 

 and isotropic. 



In the last sentence I say field, not field-entity , because 

 the terms homogeneous and isotropic refer to the relation 

 to material measuring - appliances. In fact our whole 

 argument is that the homogeneity and isotropy is not 

 inherent in the field-entity itself, but is introduced into 

 the field at the material end o£ the relation. 



The question now arises, What is this natural comparison- 

 length OQj in the field? The obvious answer is that it is 

 the radius of curvature of space in the corresponding- 

 direction. I presume that some other independent length 

 may exist associated with direction at each point of a 

 manifold ; but if it does it must involve derivatives of 

 high order, and it would seem far fetched to suppose that 

 the laws of formation of the electron are such that the size 

 depends on it. It seems clear that the natural comparison- 

 length OQi is to be identified with the radius of curvature ; 

 and accordingly the curvature of empty space is everywhere 

 homogeneous and isotropic*. 



It only remains to express this condition in the form 

 of covariant equations. It can be shown (see Appendix) 

 that the equations 



where A, is a universal constant, express the condition that 

 the radius of curvature in every direction is equal to 

 QX)~ i . These equations are Einstein's law of gravitation. 



The argument may be summed up as follows : — If the 

 electron that can exist at any place and time is on a 

 particular and not on an arbitrary scale, or, more gene- 

 rally, if the statistical average of the electrons that may be 



* This may appear to Lave some connexion with the view of 

 Dr. Whitehead that space may be Euclidean or non-Euclidean but 

 must be homogeneous throughout. But uniform and isotropic curvature 

 is by no means a sufficient condition for complete homogeneity of space, 

 and it leaves room for the full range of variation of geometry from point 

 to point required by Einstein's theory. 



