232 Dr. L. Silberstein on Boundary Difficulties 



ad hoc are not made, the difficulties or incompatibilities at 

 the surface may possibly become more acute, extending, that 

 is, even to non-homogeneous lumps of matter. 



Manifestly, the coefficient g H appearing through vJv, and 

 therefore also the i£'s, can only be obtained after an in- 

 tegration of the differential field-equations, so that in general 

 the whole problem becomes considerably more intricate. To 

 shed some light upon it let us consider the well-known 

 approximate solution embodied in the line-element 



that is to say, 



= (l+^)W + dx 2 * + d^), 



v 2 =9u=l-^ n > 



GO 



(2 a) 

 (2 b) 



where £1 is the Newtonian potential of the mass distribution p, 

 i. e. satisfying the common Laplace-Poisson equation 



The developed form of (1) is easily seen to be 



! P* 



Vi 



I^N(K-L)3-^(K-L)^^(K~L)-^|,..|=0 5 

 where 





^n v 12 v n 





9ll9l2 9u 





9n9i2 9i3 



Pz= 



92)922 929 



+ 



v 2 i v 22 v 2S 



+ 



921 922 923 





9*1 9 12 933 





931 932 933 





^31 ^32 ^33 





9n 9i2 9n 





pl= 



V 21 ^22 ^23 



+ two similar terms. 





^31 ^32 '-33 











Now, with the approximate solution (2), and taking c as 

 unit velocity, 



9u = l-2n, g 11 = 922 =g S2 =: i - (1+20) ; 

 all other g lK = 0. Rejecting higher order terms we can write 



v = l~n, \g iJ \=gn922933= : -( 1 + 6fy, 

 p 2 = (1 + 4J2) (v u + v 22 + U88 ), 

 p l= — (l + 2f2) (v 22 vm — v 2 3 + ....). 



