Relations in Einstein s Theory. 601 



tensor equation 



a -\g G=-87r*T , .... 00 



where T y is a tensor expressing the density and motion of 

 matter, k is a constant, and 



+ -v" -v log V— ^Hwl^rlog ^"^ • ( 3 ) 



in which g is the determinant of the g^ f v is the co-factor 

 of g divided by g, the Christoffel symbols are denned by 



J \~dx v cU> d^/' 



and G is the scalar <f v Gc . 



Farther, it is found that the principles of the conservation 

 of mass, energy, and momentum in their generalized form 

 are expressed by the contracted covariant derivative equation 



t;„=o (5) 



From (2) and (5) the following relations are deduced : 



^=*-sf> ® 



in which the contracted covariant derivatives are defined by 

 ~doc v 



{pv,*\ = i 



^^^-(wJG^+^^^g;, . . (7) 



where Q v lL =^Gr Ht . 



The four relations corresponding to /jl — 1, 2, 3, 4 in (6), if 

 they are true at all, must be identities derivable from (3). 

 Written out in full they are extremely complicated, and 

 Prof. Eddington's remark in his ' Space, Time, and Gravi- 

 tation ' — that he doubts "whether anyone has performed the 

 laborious task of verifying these identities by straightforward 

 algebra" — must have come as a challenge to many students 

 of the subject. However, he himself has given an algebraic 

 proof, which is not at all laborious, in the French edition of 

 his work"*. In the meantime we had obtained a proof 

 which is perhaps a little shorter than Prof. Eddington's and 

 may be easier to those English readers whose knowledge of 

 the subject is based mainly upon his " Report." In view of 



Espace, Temps et Gravitation, Complement Mathematique, p. 89. 



