806 The Relativity of Field and Matter. 



origin, and the #'s themselves have Euclidean values there ; 

 but the second derivatives contain the curvatures. 

 To calculate Gr„ v at the origin, we have 



since the remaining terms vanish with the first derivatives 

 of the g's. Using values at the origin, this further reduces 

 to 



Working out the terms in detail, we easily find 



On = -h(h + h + h), 

 G 22 = -fe(*i + *&+**), 



G44 = —^4(^1+^2+^3)? 

 G12, . . . , = 0. 

 The condition G^ = X(/ becomes at the origin 



G11 = G22 = G"33 = GU4 = — X ; Gi 2 , . . . , =0. 

 Substituting the above values of the Gr„„, this gives 

 h x = h = h = h = -v/(£X), 



showing that the radius of curvature is isotropic and equal 

 to a constant. 



The argument in the paper referred only to space- 

 dimensions, so that our introduction of the time-dimension 

 (or rather, imaginary time #4) here may seem to be going 

 beyond what has been justified. But the space-section 

 of the four-dimensional continuum can be taken in any 

 number of different directions, corresponding to the 

 Lorentz-transformation, so that actually four dimensions 

 are covered by the argument. 



