General Relativity and Einstein's Theory. 399 



homoloidal or not. Therefore, just as in the case of Minkowski's 

 homoloidal representative space, we put 

 ds^ = dr'^' + dl'\ 

 Comparing with (25), it is seen that if .^4' is identified with /', 



dr''^ = ^.g .' .dx .' dx .' , /, /=l, 2, j. 



IJ I J I J ''vJ 



Therefore, if the four dimensional representative space is 

 curved, system S' is no longer Euclidean, and points in it cannot 

 be located by means of a rectangular coordinate system. More- 

 over, the world-line of a particle at rest in S' is no longer a 

 straight line, through, since the coefficient of dl' is not a function 

 of xl, x\', x^, it is a geodesic, or curve of minimum length, like 

 the arc of a great circle on a sphere. 



If it is desired to compare a system 6^ with S' , (11) gives 

 ds"" = dr'^ -f dl'^ = h^dr^ -f k^dl\ (26) 



where, if x^ is identified with /, 



//^ dr' = fjg-jdx. dxj , i\j = i, 2, J. (27) 



gii = k\ ^14 = g.^ = ^3, = o. 



and the three dimensional geometry of S, like that of S', is non- 

 Euclidean. 



Now, just as the world-line of a freely moving particle in 

 Minkowski's homoloidal representative space is a straight line, 

 Einstein assumes that a freely moving particle describes a geodesic 

 in the warped four dimensional space which he employs. The 

 acceleration of such a particle in the geometric field equivalent to 

 a radial gravitational field is due to the fact that observations 

 are being referred to a system 6^ in which the / curves are not 

 geodesies. Relative to this system the world-line of a freely 

 moving particle is given by 



^y^^ |^^/y"''"/^^V" "" ' '''^'" ^' ^' ^' ^- ^'^^ 



where the ^'s, like Ji and k in the simple case treated at the end 

 of section (a), may be considered as generalized potentials de- 

 termining the nature of the geometric field. 



For reasons already stated, the g's must not be functions of /. 

 Even so, they cannot be determined unless further restricting 

 conditions are introduced. These necessary limitations on their 



