398 Leigh Page, 



provided the straight or homoloidaP representative space here- 

 tofore employed is replaced by a curved four dimensional surface 

 in homoloidal space of higher dimensions. 



It has already been noted that physical measurements are con- 

 fined to the observation of intersections of world-lines. All that 

 experimental science can tell is the order in which these inter- 

 sections occur. There is no a priori reason that the results of 

 experiment can be more conveniently represented by plotting 

 world-lines in a homoloidal four dimensional space than in a 

 curved or warped space. Therefore when Einstein found that 

 the equivalence hypothesis could not be applied to a radial gravi- 

 tational field so long as Minkowski's homoloidal space was re- 

 tained, he was entirely justified in replacing this straight repre- 

 sentative space by a four dimensional space which is warped in 

 homoloidal space of higher dimensions. 



On a curved surface it is not generally possible to use rectangu- 

 lar coordinates, or any other system of coordinates, such as polar 

 coordinates, which are derivable from rectangular coordinates 

 by a mathematical transformation. Therefore the linear element 

 ds must be given in terms of a set of curvilinear coordinates a\, 

 Xo, A\, x^ by an expression of the form 



ds^ = ^.p- :.(/x .dx ., (^..= p-.., /, /'--/, 2, J, 4, (25) 



where the ^'s may be functions of the x's. For example, in 

 locating points on a two dimensional spherical surface of radius 

 R, such as that of the earth, it is necessary to use coordinates 

 such as the co-latitude 6 and longitude ^. In terms of these the 

 linear element of the spherical surface has the form 



ds\= R^d 0^ + 7^2 sin-'e d <p\ 

 for which case it appears that 



dx^ = d 9, dx^ = (/ <f>, 



gii = R^ go. = R^sin-e, g^^^ — g„^^ = o. 



Now transformation (11) between a system S' characterized 

 by the constant light speed c and any other system 6" must hold 

 no matter whether the four dimensional representative space is 



'The term homoloidal as applied to space of four or more dimensions 

 has the same significance as the word Euclidean applied to three dimen- 

 sional space. 



