1083 



The integral on the lefthand side, which multiplied by 4.t evidently 



gives the space-integral |^^ — g G dV over an empty spherical space 



V, has been calculated by Droste. He used polar coordinates, but, 

 the integral multiplied by 4.t [t^ — t^) giving a scalar, the result is 

 independent of the choice of the space-coordinates. First Droste 

 finds for G, which evidently is also a scalar, (see Droste, "Het 

 zwaartekrachtsveld" p. J 6) 



^22 r'* 4 v'w 4 v" 4 u'v' 2 iv" 2 u'w' 



G = 1 H , . (29) 



V II V icviü ir V ,u V u w u ic 



where u' v' lo' are derivatives with respect to r. Further Droste 

 finds: 



(30) 



ƒ;. ^, r\ d rvhc'4-2vuw'\ wv'" -\-2vv'ic ) 



|/_^ Gr'dr=2 - v;( ) H ~ h ""' dr. 



All variations being taken zero at the limits ?' = ?\ and r = 7^,, 

 we have 



r. / ^ riwv" -f- 2vv'w' ) 



(f V—g G r- dr=2(f I \- im dr 



'1 '-1 



This is now our expression for the lefthand side of equation (28). 

 Now we must consider the righthand side of this equation, and we 

 shall begin by proving the following relations : 



_ d^^-i da--^ 2 ^ ^ 

 ,„.v 0^-"^ Ou u '■ 



'^^— 4- = — ^/'> • . . . . (32) 



„ diO? dö/^^ 2 



vv^here <,r^ and Jy are connected with the tensor 5; in the way 

 indicated by the equations (27). In order to prove the vaHdity of 

 the equations (32), we first remark that because of the spherical 

 symmetry both the lefthand and the righthand side depend on r 

 only. If the equations hold for an arbitrary point on the AVaxis 

 {x^ = r, A', = x^ = 0), they are always valid. 



According to (26) and (27) we have for points on the Z,-axis : 



I 1 ' r' 1 



9'' = —,. g"=g" = --^ = --, g''=~. (33) 



U p^ V w 



3:/ = J/, 3:,* = i,»=^i; (34) 



