25 



só that of the stress-energj-componenls of the matter only one is 

 different from zero, namely 



Further (66) involves that, also of the quantities Tab, only one. 

 namely 1\„ is not equal to zero. As we may put ^^ — (/ = cr\ we 

 have namely 



Finally we are led to the three differential equations 



A + 2 r i' + i rU" - fi - hr (i' -V- ^ r v' = — \ x q, . (104) 

 2r?J -\- r' r' — rfx' + ^r'v" = - ^XQ, . • • (105) 



^V' + l,.»X,"r=: iXi>. (106) 



It may be remarked that gdx^ch^ch, represents the "mass" present 

 in the element of volume clv.cLv^dx,. Because of the meaning of 

 x„x^,x, (§ 48) the mass in the shell between spheres with radii 

 r and r + dr is found when Qd.i\dx,d.v^ is integrated with respect to 

 a\ between the limits — 1 and + 1 and with respect to .i-, between 

 and 2.T. As q depends on ?• only, this latter mass becomes 4.riQdr, 

 so that Q is connected with the "density" in the ordinary sense of the 

 word, which will be called q, by the equation 



The differential equations also hold outside the sphere if q is put 

 equal to zero. We can first imagine (j to change gradually to 

 near the surface and then treat the abrupt change as a limiting case. 



In all the preceding considerations we have tacitly supposed the 

 second derivatives of the quantities (/„è to have everywhere finite values. 

 Therefore v and v' will be continuous» at the surface, even in the 

 case of an abrupt change. 



§ 58. Equation (106) gives 



r r 



v' = ~ ( Qdr, ...... . (107) 



r 

 



where the integration constant is determined by the consideration that 

 for r = all the quantities g„b and their derivatives must be finite, 

 so that for r == the product r^r' must be zero. As it is natural to 

 suppose that at an infinite distance v vanishes, we find further 



