236 Boundary Difficulties of Einstein s Gravitation Theory. 



Now, let the body be a homogeneous sphere of radius R, 

 and of density p ; or let it consist of concentric strata, and 

 let p be the density just below the surface. Then we shall 

 have, at the inner side of the surface r=M, 



*rf.-f + »(F + J)l 



}>,. . . (8m) 



giving Ki+Ki + Kz^F 2 , and at its exterior side (/> = 0), 



k 1= k 2 =f(f + i)] 



V ' J., . . . (8«,0 



*—*(*+h) j 



making JKi + JT, + JT 8 == J? 1 + }«p. Thus, if F= dajdr is con- 

 tinuous across the surface, so is K n =K ni the principal cur- 

 vature corresponding to the radial or normal axis, and there 

 is thus, for the sphere under consideration, no geometrical 

 incompatibility. 



But (to repeat it) whether such will also be the case for 

 differently shaped bodies seems doubtful. Levi-Civita, in a 

 private letter to the writer, expresses the opinion that the 

 continuity of the normal principal curvature K n (as the 

 above K 3 ) will also in general be ensured automatically in 

 virtue of a certain equation given in treatises on differential 

 oeometry. (Levi-Civita quotes Bianchi's " Lezioni di geo- 

 metria differentiate" vol.i. p. 373, unfortunately not accessible 

 for the present). But I do not see how such an equation 

 can hold independently of the details of mass distribution' 

 and therefore of the distribution of the values of £2 and of 

 its derivatives. The reader may find it worth his while to 

 investigate on similar lines as our above sphere the case of, 

 let us say, a homogeneous ellipsoid of revolution. 



As I have attempted to point out in another paper (Phil. 

 Mag. vol. xxxvi. p. 94) there are grave objections against 

 Einstein's gravitation theory of an entirely different nature. 

 Until these are removed, it would seem useless to insist any 

 further upon the purely geometrical difficulties. 



London, 



December 28, 1918. 



