1089 



the meaning we ascribed to them, is evident from the consideration 

 of a point on one of the axes of coordinates. 



We shall now transform formula (41). In the original fonr- 

 diraensional system of coordinates this is after a slight variation 



(/ r' p* w w' 



r'x W. 



dr V_-g 



Now we shall prove that the lefthand side remains covariant at 

 the transformation of the time-coordinate. As also the righthand side 

 remains invariant, the formula holds in this form also in the four- 

 dimensional system of coordinates. According to (40) we have for 

 every system of coordinates 



7 being a mixed volume-tensor, J/ is transformed according to 

 this formula : 



V — q dx. da?a — ., 



V = ^-=I^T^^ 1/ («) 



If we consider a point on the AVaxis, tlien clc^=^dr. kX our 

 transformation of the time-coordinate ^ = 1 is the only one of all 



- — which is not zero- Of all ^— only ~z^ = 1 and -^^ = i|? are 

 different from zero. As further g = g we find 



It T/ was not zero this would mean that there existed a radial 

 energy-current and the energy of the system would change contin- 

 ually. As we assumed the field to be stationary, we have j/^O 

 and therefore 5/ = j/. As ^l a"- is a volume-scalar and as the 



determinant </ does not change by the transformation, ^ï^/ does not 



undergo a change by the transformation either. Thus at the transfor- 

 mation ¥^ remains invariant. 



As r, p, w too remain constant, we thus obtain for the new 

 four-dimensional system of coordinates also 



r'x Vrrr— -^ —- (41a) 



dr \\/—g dr J 



