234 Dr. L. Silberstein on Boundary Difficulties 



calculated as the roots of (4) and then tested for their 

 compatibility or incompatibility at the prescribed surface of 

 the body. Let K n stand for the Kiemannian curvature near 

 the boundary S of the body corresponding- to a geodesic 

 surface element oriented tangentially to S. Then if K n is 

 continuous across the boundary S, there will be no incom- 

 patibility. But if K n is discontinuous, then, no matter 

 whether the body is homogeneous or not, certain forms of $ 

 as boundary will obviously be excluded, and will — if forced 

 upon the body — lead to contradictions. Now it seems very 

 doubtful that, with the above determination of O, the curva- 

 ture K n should be generally continuous across the boundary, 

 especially as K n contains also the second derivatives of the 

 potential. And, for the same reason, it seems doubtful 

 whether the usual conditions of the integral of V 2 ^= —\ K P 

 can be successfully replaced once and for ever by some other 

 boundary conditions. The circumstance that with the 

 actually existing densities the amount of discrepancy may be 

 only small * and even negligible for the physicist, does not 

 change the position. For the question is one of principle, 

 and would certainly call for a thorough mathematical 

 investigation by the strict adherents of Einstein's theory. 



No solution of this difficult general problem will be 

 attempted here simply because it is beyond the powers of the 

 present writer (and quite apart from his disinclination to 

 Einstein's theory based on entirely different grounds). In 

 order, however, to explain the above general remarks and 

 formulae, it may be well to give here the solution of our 

 cubic for the simplest case of a spherical body, for which (as 

 was to be expected) there is no geometrical incompatibility. 



Let r be the distance from the centre of the sphere and R 

 its radius. We might with equal ease treat the case of p 

 equal to any function of r. But to fix the ideas, let the 

 sphere be homogeneous. Then, a being a constant pro- 

 portional to its total mass, we have 



fl= - outside, and H = w „r 2 inside. 

 r E 3 



Thus, writing F for the gradient V^, and 



dr 

 for its absolute value, and using the abbreviation 



dr r 



* The radius of curvature corresponding to %ko is for water of the 

 order of 25 astronomical units. 



