General Relativity and Einstein's Theory. 407 



Starting" with the vector P , we may form a covariant dyadic 



by applying (52), and by operating on this dyadic with (53) 

 a covariant triadic is obtained. This triadic, which we shall 

 denote by A, will contain first and second derivatives, and be 

 linear in the latter. Evidently 



A =(D - -^v, /d} • l'^- -\crv,T\ -l"^-) (D -l(r/x, e! ■ f ^V 



= D' P - >v, p[ ■ D P - io-v. tI • D P — Jo-^, e- • D P 



+ {/Ai/.p;-jo-p. e;- p^+;o-v,t;-;t/a, e| • p^- d^-o-/^, ei • p^. 



Now interchange of ju, and v will have no effect on the first 

 . five terms. Therefore 



\p.<T ~ ^fXV<T^ ( ' ^''' ^ ' ■ ' '■^' ' ' ~ ' "■/"' ■"'■■' '■''' ' '■ + 



D^lcrv,e:-D^!o-/x,e;) • P^ , 

 and 



B^j^^=-o-v, Ti-;T^, e;-|o-/i, ri-JTV, ef+D Jo-v, ej— Dj^'or^-ef (54) 



is a tetradic which is partly covariant and partly contravariant, 

 as indicated by the suffices. Obviously, if the four dimensional 

 representative space is homoloidal, the ^'s defining the line ele- 

 ment may all have the constant value unity. Therefore, as every 

 term in this polyadic contains a derivative of one of the ^'s, it 

 will reduce to zero in this case. Conversely, it may be shown that 

 if this polyadic vanishes, the representative space is homoloidal. 

 Therefore the equation obtained by equating B to zero cannot 

 represent the law of gravitation. 



A less stringent equation is obtained by equating to zero the 

 dyadic formed from B by putting a dot between k and k . 



This equation, i. e. 



^iirr '" '^»'''^J : {-rp-, v\ — {(tii,t\'{tv ; v] +-D \cTv; v\ —D ■ {cr/A, v} 



= (55) 



is taken by Einstein for the law of gravitation. Evidently it 

 satisfies all the necessary conditions, and it is probably the simplest 

 equation which does. 



Equation (55) may be somewhat simplified. For 



.Av 



