1239 



We put 



y. = y^ >^' — Q- 



(p indicates then the electro-static potential, q the density of the 

 electricity. Further we assume, that the field possesses spherical 

 symmetry and choose the tiine coordinate so that (/,4Z=0. We then 

 have y —g :=zzuto p-, and for ^:)3ï(«> we find the following expression, 

 the validity of which is seen most simply for a point on one of the 

 axes of coordinates, but which must be generally valid as iHï'^' 

 depends on r only and not on the direction from the centre, 



9»(«) = - /^ f^'Y + 2,M. (8) 



4jr mo yar J 



For the integral of ?)ï(«' over a 4 dimensional extension of a titly 

 chosen form we obtain 



( M pK«) dx, dx^ dx^ dx^ =z= 4.T {t,-Q \n^') r' dr, . . . (4) 

 where 



'2 '1 



J/- I r' n" /"du'Y ) 



SW(«) r" dr= I } — --^ J 4- 8 JT (/> () r' dr . (4a) 



The laws for the electric field we found by variation of 7, while 

 u,n),p,Q were kept constant. As the expressions for©* and ^^^("'' do 

 not contain (p, we obtain by this variation 



dr \ uw dr J 



The integration gives 



- r^p' dip d<p nw e{r) 



-T- = e (r), - - = -^ (Ö) 



mo dr dr p r 



where e{r) denotes the total chai-ge iu a sphci-e with radius r. 

 Outside the body eir) is constant and equal to the total charge 

 of the body. 



As in the infinite in has the value c (see 1 p. 1079), we easily 

 see that e and <> are its charge and density in electro-magnetic units, 

 (p on the contrary the potential iu electrostatic units. 



Now we shall calculate the gravitation field. This calculation can 

 be made with the aid of formulae (38)1. Then we must first calcu- 

 late the stress-energy-tensor for the electro-magnetic field by replacing 

 ?W in formula (2) I by ^^K^) and by introducing the expression (2j. 



