4o6 Leigh Page, 



and D G' ^ 



are written down, and if (48) is subtracted from the sum of 

 these two, it is found that 



i(D' G\+D' G'.-D\G' '\ = G p : D'f D''"" 

 + i (D Gn +'DoG -D G n| ; D'l d" D'^. (49) 



'^ \ a py p ay y ap ) A /a v \^^/ 



Forming the dot product of G' ^- D'' and (49) it follows that 



r 



iG'^^ • (D' G '. + D' G' X- D'x G' ) • d' = d'" 



\ /X VA V fXA A fXVl p VjJL 



+ iG'^^ • (D Go +DoG - D G o ) : D''^ D'^. 



' \ a py p ay y ap I [x y 



Christofifel's triadic is defined by 



Hence 



1/xv, p]' • D'^ ■ P = D'"' • P + ;a/?, el : D'" D'^- P , 



" '^ ' p € vfx e ' ' fx V e 



O'* l/xv, p\' ■ P' = D''' • P +D'' D'^: {afi, e; • p . (50) 



Subtracting this equation from (47), 



D' P' -\^v,p\'-p' =D'"d'^:/DoP -;a/3,e|-p ), (51) 

 V jx '^ ' p u V \ p a '' ' cl ^'^ ' 



showing that the dyadic on the left hand side is covariant. 



Therefore the expression 



d^-|/xv,p!-i^. (52) 



is a covariant differential operator when applied to a vector P . 



Now consider the triadic 



Q (D P -■j/ttv,p;- P 1+P (D Q -{<n',T\ Q \ 



= D ( P Q ) — >v. /3 ! ■ P Q — ; o-v. T I • Q P . 

 v\ p- cr ' p^(T " ^r p. 



As it is made up of sums of products of covariant vectors and 

 dyadics, it is covariant. Hence the differential operator 



U-{pv.p]-lf^--{av,r\-f. (53) 



produces a covariant triadic when applied to the dyad P Q . But 



a dyadic is merely the sum of a number of dyads. Hence this 

 differential operator forms a covariant triadic when it acts on any 

 covariant dyadic. 



