69 



we do not need (his). The system of coordinates will be fixed by the 

 conditions : 



V = r, viz. p = r (3) 



Putting furthei": 



1 = )/—g T = uwT, (4) 



and applying a simple transformation, we can write for the mentioned 

 starting formulae: 



1 



10 



-- l4-2r- +l=zr'xT;, .... (5) 



d 

 dr 



1 -71 \=^''^n^ ...... (6) 



2 ], ,• W A ,. dTr 



-{T';-T\) ^-{T\-T\) = -1 (7) 



r w dr 



These formulae hold for each stationary gravitation field with 

 spherical symmetry ; the system of coordinates only is determined 

 by the condition (3). Tlie quantities u and lo determine (when 

 /) = 1) all components <j,j,j of (he fundamental tensor according to 

 the formulae i25) I. 



When T/ is given, the equation (6) determines u as a function 

 of r. By integration across a layer which afterwards by a passage 

 to the limit is changed into a surface of discontinuity with 

 radius }\ = i\ ^ R and after division by R we obtain 



'a 

 z= Rk Urn j ^1 dr (8) 



This formula shows that u changes discontinuously at a surface 

 of discontinuity where 



Urn I Ty 



dr 



differs from zero. Such a surface which moreover satisfies the condition 

 (2) will be called a material surface. The system of coordinates might 

 be chosen in such a way that at the surface u changes continuously, 

 but then /; would change discontinuously. In general at least one of 

 the space-components of the fundamental tensor changes discontinuously 

 at a material surface. With the aid of formula (5) we shall now 

 prove, that w on the contrary changes continuously at our material 

 surface, when only the condidon (2) is satisfied. Equation (5) gives 



to' U' { r\ \ 



2- = - l-r'x7'; , ..... (9) 



vr r V J r 



