71 

 We thus obtain 



\-W>iT'r r] 1 rdu u^-nj^ 1 ^ ,\ 



hm I = I du — i — — ^1 1 — R* K 7', 



Now we have treated one part of the left-hand side of (li) by 

 integration and by passage to tlie limit, (^f the remaining parts of 

 this left-hand side those containing 7','. remain zero at tlie passage 

 to the limit according to our assumption (2), u remaining moreover 

 finite. The part containing 7'j; on the contrary does not become 

 zero. The right-hand side has the value zero at the limit, as we 

 have assumed 7'|." to change continuously at the surface of discon- 

 tinuity. Multiplying our equation still by iv, which quantity we 

 have proved to change continuously at the surface, so that at the 

 limit it may be considered as constant, we obtain : 



''=-£e^"'-"K'-^,- "'"''"') • • ■ <'^' 



Together with (8) this formula expresses 'the laws for a surface 

 of discontinuity of the kind we now consider. Tiiese formnlae will 

 be applied to the special case that all matter that is present is 

 situated in the material surface. 7", being continuous, we have in 

 this case 7" =: 0. Further we have according to (6) both inside and 

 outside the surface 



1 ) = const. {r^\=R} .... (13) 



When r = 0, u cannot be zero, so that the value of the constant 

 within tlie surface must be zero. We thus find for r <^ R, u =z 1 

 and therefore also 



'^. = 1 (14) 



Within the spherical material surface we thus have a euclidic 

 space. (This is of course true for every hollow sphere; the distri- 

 bution of mass and stress on the outside only has spherical symmetry). 

 Outside the material surface the constant in equation (13) has not 

 the value zero, but a value, proportional to the mass of the system 

 which is given by formula (15) II: 



4Lnn 

 m = . . (15a) 



We thus have for u, -. 



