1085 



field with splierical sjmmetry. We can easily deduce from il tlie 

 following equation 



u' v' w' \ d 



-X/+2-5/ + -r/ Uz-Cr'X/), . . . (39) 



u V '^ w J dr 



which can also be found immediately bv applying formula (22) of 

 EiNSTKiN (Hamiltonsches Prinzip) to our case. Formula (39) ex|)resses 

 that the spherically symmetrical material system is in equilibrium 

 when the gravitation is taken into colisideration ^). 



Starting from equation (18) we shall now deduce a formula for 

 the energy and tlie mass of the system. We put 



^ = t,' -I,' -l^-l,' = l: -Z/ — 21/ . . (40) 

 and calculate r'xV'. Putting for r'y.^^\ /'"x j/, r^y. IP the expres- 

 sions following from (38), we find that most terms neutralize each 

 other and we obtain 



WW V w Zv wu 



r^aW =A h 2 , 



u u u^ 



d Cv^ 10 \ d / r" p^ iv'\ 



r^^W=2-( ] = 2-(-J^-] .... (41) 



dr \ u J dr \ ti J 



Outside the material system is V^O and we thus have for r^ i? 

 {R being the radius of the body) 



2r» ?-^ — constant {r\> R) . . . . . (42) 



u 



The meaning of the constant will be examined later on. 

 Equation (41) suggests a connexion with our former equation (20) 

 and we shall directly see that this really exists. 



Excluding the theoretically possible case that is oo at the centre 



u 



of the system we find by integration of (41) from ?• = to an 



arbitrary upper limit r 



P 



Ir' X Wdr = 2 r* 







1) If we put 



u ^ 10 =: 1 , V ^= r , 

 viz. if we neglect the gravitation (39) becomes 



2r3:/ = ^(r'r/), 



which equation expresses the equilibrium between the ponderomotoric forces given 

 by the stress-tensor Ï fur the case that there is no gravitation. 



75 

 Proceedings Royal Acad. Amsterdam. Vol. XX 



