602 Dr. Gr. B. Jeffery on the Identical 



the fundamental importance of these identities in the theory 

 of Relativity, we venture to put forward this proof. 



It is known that a transformation of coordinates may be 

 found in an infinite number of ways so that at one definite 

 point of the continuum the g v and their first differential 

 coefficients have prescribed values. Select a definite point 

 of the continuum which we will call the origin, and if 

 necessary make a transformation of .coordinates, so that the 

 first differential coefficients of the q all vanish at the 

 origin. The differential coefficients of the second and higher 

 orders will then in general not vanish at the origin. We 

 will first show that in the new coordinates the relations (6) 

 are identically satisfied at the origin. 



In the case when the first differential coefficients of ihe 

 g vanish, the Christoffel symbols also vanish at the origin, 

 and we have from (7) 



9 dx v ' 



Substituting from (3) and omitting terms which obviously 

 vanish on differentiation, we have 



Q» _ _1,M n °P <^ /B#A0 , d,<7 M /3 B^V\ 



Kv 



v 



+ g — log y - q. 



The second and third terms in the bracket cancel on 

 summation, and we have 



<*---*> * ^»,,+^ 9^^.v logV -^ (8) 



Again, 





The first and second terms are identical on summation ; 



