26 



v = x I ^ I Qdr (108) 



The quantities l and n on the contrary are not completely determined 

 by the differential equations. If namely equations (105) and (106) are 

 added to (104) after having been multiplied by — \ and -j- ^ respecti- 

 vely, we find 



;. + rl' — n ^ rv = ..... . (109) 



and it is clear that (104) and (105) are satisfied as soon as this is 

 the case with this condition (109) and with (106). So we have only to 

 attend to (108) and (109). The indefiniteness remaining in A and fi is 

 inevitable on account of the covariancy of the field equations. It does 

 not give rise to any difficulties. 



Equation (107) teaches us that near the centre • 



if Qg is the density at the centre, whereas from (108) we find a 

 finite value for v itself. This confirms what has been said above 

 about the values at the centre. We shall assume that at that point 

 P., (Ji and their derivatives have likewise finite values. Moreover we 

 suppose (and this agrees with (109)) that )., fx, )J and (i are 

 continuous at the surface of the sphere. 



If a is the radius of the sphere we find from (108) for an exter- 

 nal point 



•J' 



I' :=: I O dr, 



O 



Without contradicting (109) we may assume that at a great 

 distance from the centred and (x are likewise proportional to - , so that ?.' 



r 



and ii' decrease proportionaUy to -. 



§ 59. We can now continue the calculation of t'/ (§ ^6). 

 Substituting (101) in (99) and using polar coordinates we find 



Ay. V V \^ u' uw J 

 whence by substituting (102) we derive for a field without gravitation 



This equation shows that, working with polar coordinates, we 



