74 



of tlie electric sphere. As was proved in II § 1 we have outside 

 the sphere 



uto^c {r^ R), ...... . (23) 



T:.= tI = -^^ (ryR) .... . . (24) 



As inside the sphere and at its surface ii=:l, we find from (6) 

 by integration up to an upper limit r ^ R 



/ 1\ r 4 xe* He' 



\ w'y J OJT r öjr It 



R 



1 xe' xe' _^, 



- = 1 \ (25) 



A comparison with equation (11) II shows, that we must have : 



xe" 



SjtR' 

 and {15a) gives for the mass m 



m= — ......... (26) 



2R ^ ' 



The charge e being expressed in electro-magnetic units (see II 

 p. 1202) this expression for ?/? is equal to the electro-static energy 

 divided bj c^ Besides the electro-static energy no energy occurs in 

 our system. That outside the electric body no gravitation energy, is 

 present has been proved already in II § 2. The last result says 

 therefore that neither in the electric surface any gravitation energy 

 is accumulated. 



§ 4. A sphere of an incompressible fluid. 



This problem has been treated already by Schwarzschild, ') but 

 as the formulae (5), (6), and (7) lead us by another way quickly 

 to the same result, it may be allowed to develop these calculations 

 as shortly as possible. 



That the medium is incompressible means that when at rest 



t\=q (27) 



is a constant characteristic for the medium. The fluid character of 

 the medium demands further that no tangential stresses can occur, 

 so that we have 



rr = Tl = -p (28) 



^) K. Schwarzschild, Ueber das Gravitationsfeld einer Kugel aiis inkompressibeler 

 Fliissigkeit nach der Einsteinsclien Theorie Berl. Ber. 1916 p. 424. 



