of Einstein s Gravitation Theory. 231 



tares K^ K 2 , K- 6 of the three-space (x u <r 2 , a? 3 ) within a lamp 

 of matter 



det'(7f-Z)^-/,7^,|=0, .... (A) 



where L=k(^T 4a —^T)+\ and det stands for the deter- 

 minant of the nine elements (ij=zl, 2, 3). All 3T.. (stresses, 

 &c.) being negligible with the exception of T U) so that also 

 T=r/ u T u = p (density), the above cubic became 



(K-L) 3 .det % |=0, . . . . (B) 

 and gave at once 



K 1 =K 3 =K 3 =iA: / o+X, 



i. e. an isotropic space, thus corroborating the original 

 "bizarre" conclusion according to which " a homogeneous 

 body could have only the shape of a sphere." 



In deducing equation (A) I have based myself, among 

 other things, upon certain differentially geometrical relations, 

 due to Ricci, given in a paper by Levi-Civita (Atti Lincei, 

 vol. xxvi. p. 641). Now, Prof. Levi-Civita, to whom I have 

 communicated the above and some previous results, has 

 recently (in a private letter) called my attention to the fact 

 that the G-- of his paper just quoted are not identical with 

 mv ^ip Du k differ from them by 



where v = vg u and v.- is the co variant derivative of v with 

 respect to x { . x ., corresponding to the three-space element in 

 question. 



Thus, neglecting the stresses, as before, the so-called 

 coefficients of Ricci become 



and we have, instead of the above (B), the following cubic 

 det|(K-%..-i,..|=0, . . . . (1) 



with L = ^Kp + \, which gives in general three different curva- 

 tures K l9 iT 2 , AT 3 , depending essentially upon </ u or v 2 . Thus 

 the space is no more isotropic, and the original conclusion 

 about the shape of a homogeneous body ceases to be necessary. 

 But the geometrical boundary difficulties do not vanish 

 therefore ; on the contrary, unless some boundary conditions 



