General Relativity and Einstein's Theory. 405 



As ds^ is an invariant, the right hand member of this equation 

 is also an invariant. But it is evident that if the scalar product 

 of two dyadics is an invariant, and one of these dyadics is con- 

 travariant, the other must be covariant. Hence as 



(/p dp'' 



ntravariant, G o must 1 



ap 



If 



is contravariant, G o must be covariant. 



ap 



y--S 



is the reciprocal of G o, then 



and 



G = 1, 



ap 



^a^-^' 



showing that G is a contravariant dyadic. 



In ordinary vector analysis a covariant dyadic may be con- 

 structed from a covariant vector P by forming the open product 

 V P. When the ^'s are functions of the coordinates, however, 

 a dyadic so constructed would not be covariant. For it follows 

 from (40) that 



P.. (47) 



the presence of the second derivative on the right hand side 

 destroying the covariance of the dyadic. The value of the term 

 involving the second derivative may be found by constructing 

 and combining triadics as follows : 



^' G' =D'«D'^: G ., 



pv p. V ap 



D\G' =G o:(D'^D'f+D'^D'f)+D'lD'"D'^;D G . (48) 

 A p.v ap \ V Ap. p. Av/ A p. V y ap ^ ' 



in which some of the dummy suffices have been replaced by other 

 letters, and use has been made of the fact that 



/3a a(i ' 



i. e. G o is self conjugate. If similar expressions for 



D' G' . 



W VA 



