400 Leigh Page, 



values are evidently expressible by means of a set of equations 

 containing the g's and their derivatives. Now these equations 

 have a two-fold significance. In the first place, the ^'s determine 

 the nature of the curved four dimensional representative surface 

 in homoloidal space of higher dimensions. Therefore the equa- 

 tions relating them specify certain absolute properties of this four 

 dimensional space which are independent of the particular refer- 

 ence frame to which physical observations may be referred. 



Viewed from another aspect, the ^'s have a rather different 

 significance. It has already been pointed out that they are the 

 potentials of the gravitational field which they specify. Conse- 

 quently the differential relations connecting them may be inter- 

 preted as the law of gravitation of the field. Now, the principle 

 of general relativity requires that this law shall differ when 

 referred to different reference systems only in so far as the 

 geometries and devices for measuring time and space together 

 with their units may differ in these systems. But the metrical 

 properties of a system are defined by the values of the ^'s. Hence 

 the law of gravitation must have the same form in terms of the 

 g's and x's of one system 5" as it has in terms of the |:"s and .r"s 

 of any other system S'. 



Furthermore, the law of gravitation must reduce to Laplace's 

 equation as a first approximation. Reference to (24) shows that 

 this equation has the form 



which suggests that as a first approximation 



where r is the radius vector and m is a constant proportional to 

 the mass of the attracting body. The small quantity - is evi- 

 dently a measure of the divergence of the geometry of the four 

 dimensional representative space in the neighborhood of an 

 attracting mass from the homoloidal geometry of this space in a 

 region where no matter is present. 



(c) THE LAW OF GRAVITATION, 



In seeking the law of gravitation, the following requirements 

 serve as guide. First, the law must be a differential relation 



