General Relativity and Ei)iste{)i's Theory. 397 



The problem to be solved consists in finding a geometric field 

 which (a) conforms to Newton's law of gravitation as a first 

 approximation, and (b) has the properties of a Euclidean system 

 with constant light velocity c at great distances from gravitating 

 matter. As the properties of a permanent gravitational field do 

 not change with the time, the values of Ji and k in the expression 

 for the linear element in terms of the coordinates and time of the 

 equivalent geometric field must not involve /. But it has been 

 shown that if h and k are not functions of /^ the only possible 

 form which the linear element can take, (other than (12) ), is that 

 given by (i/). The acceleration of a body moving through 

 this geometric field is given by (19). If the velocity of light 

 is constant, this expression gives a constant acceleration (for any 

 given value of the velocity). Consequently this geometric field 

 may simulate a uniform gravitational field in so far as the motion 

 through it of material bodies is concerned. 



The consideration of radial fields, however, brings to light 

 serious difficulties. For (19) approximates the acceleration in a 

 radial gravitational field only if the velocity of light varies 

 inversely with the square of the radius vector. If such were the 

 case, however, the properties of the geometric field would fail 

 to approach those of a Euclidean system with constant light 

 velocity c at great distances from gravitating matter. Hence, 

 under the limitations imposed by the nature of the X Y Z L 

 space, there exists no geometric field which approximates a 

 radial gravitational field. 



The question arises as to hov/ the nature of the four dimensional 

 representative space may be modified so as to impose less stringent 

 restraints. Consider for a moment the X L plane. The form of 

 the linear element defined by (14) shows that the fanfilies of 

 curves 



X =z const, 



/ = const, 

 must be orthogonal. Now the number of mutually orthogonal 

 families of curves on a plane is strictly limited. If the plane, 

 however, may be replaced by a curved surface in three dimen- 

 sional space, much less restraint is imposed upon the character of 

 orthogonal families of curves on the surface. Such consider- 

 ations suggest that the desired geometric field may be found, 



