of Einstein s Gravitation Theory. 233 



Thus, writing for the sake o£ shortness 



o-=K-L=K-^~.\, 



or, since (in the case of water, for instance) \ is more than 

 10 10 smaller than ^, 



a=K — j>fcp, (3) 



the cubic (1) becomes, after the rejection of terms such as 

 2X2, 612 in the presence of unity, 



< 7 3 -a 2 (n n + O 22 + I2 33 )H-o-(I2 11 I2 22 --r2i 2 + &c.)--i%|=0,(4) 



where 12.- are co variant derivatives of the Newtonian 



v 

 potential O (with respect to x^ aj) corresponding to the 



three-space element (2 a). In this equation, generally 

 speaking, no term can be neglected, and we have, therefore, 

 three different principal curvatures A l5 K 2 , K z . 



The covariant derivatives of the potential appearing in 



(4) are 



where the three- index symbols are to be taken with respect 

 to the line-element (2 a), i. e. for the tensor 



1 + 212, 0, 0, 



0, 1 + 212, 0, 

 0, 0, 1 + 212. 



This gives, with the above degree of approximation, 



Q n = 



cUY 



(a*i/ + UJ Ha J 



12 B^iB^ 2 **a#i * d# 2 J 



(5) 



with similar expressions for X2 22 , 33 , and I2 23 , 12 31 . Thus the 

 coefficient of a 2 , for instance, assumes the elegant form 



n 11 +I2 22 + a 3 3 = V 2 12 + (Vn) 2 , . . (5 a) 



where V is the Hamiltonian or, in our present case, the 

 gradient. Similarly the remaining two coefficients of the 

 cubic (4) will be determined by (5) in terms of the first and 

 the second derivatives of the Newtonian potential 12 of the 

 given body (mass distribution). Thus, if 12 is found with, 

 say, the usual conditions of continuity and those "at 

 infinity," the curvatures inside and outside the body can be 



