70 



and by integration across the layer we obtain 



r\n'' r 1 I 



log w,' — log 10,' = \ {-{l-r'nTr) \ dr . . . (10) 



J { r r \ 



We slmll only consider gravitation fields in which u is every where 

 finite and when in the limit we pass to an infinitely thin layer the 

 limiting valne of the integral on the right-hand side becomes zero 

 according to the assumption (2). 



Now we shall apply formula (7) and substitute in it the expression (9) 



for — and the expression (6) for T^\ Multiplying further by 



to 



urdr 



-, we find 



u {iyT'r)dr-\-\{u\\-~r'xT\)-u\ - -W 1- - ]-tr\dr=--~^ dr. (11) 



[ xr dr \ xr J ) 2 dr 



This equation must be integrated over a layer and afterwards 

 we must pass to the case of an infinitesimal depth. In order to 

 obtain as a first term on the left-hand side the surface tension P 

 as defined by equation (1) we must moreover multiply by m. We 

 shall however not continue our general investigation, but rather 

 consider two more special cases. 



^ 2. Investigation of the state at a material surface. 



First we inxestigate the case that at the limit 7V surpasses any 

 value, so that the right-hand side does not become zero, but that 



dTl 



— -- remains finite, so that on both sides of the surface of discontinuity 

 dr 



7"! has the same value. 



In (11) we first consider the part of the left-hand side which 



after integration gives 



=*>' 



1 (/ / 1 



(l-r« HT,)-u\ — ~r(\~~]dr = 

 jcr" dr V u 



• 1 



(1 v'TiT'r) - n\d\r[\ 



4>t,7 r" ' V "' 



We have to calculate the value of this expression for the limit 

 7', — ?', = 0. In this limiting case r constant =. r^ = r, =: R, so that 

 we have 



