29G 



WILSON AND MOORE. 



16. Covariant differentiation of a simple system. Let us 

 now substitute from (33) for the second derivatives in (31). Then 



^^' = ^kHikyn ^^ + S.(Zfc)2,,c«(a('=») 



dxj 



dyi 



13 



— ^k{Xlc)^rsiyiry 



/.(a(^^)) [Y] 



k I 



L P J 



Now the presence of the multipliers jik, Jn on^the [right makes it 

 look as though the left might be a covariant of the second order and 

 if we replace (A^^) by its value l^mXmCmk and (a^^^^ by its value 

 Spga^P'^YpjtTg^, we find that 



MXk)^uCti{a'''^) 



ij 

 t 



= '^ktlmpqXmCmkCtiypkygia^^'^^ 



= SpgZpa(p«) 



ij 

 t 



1 J 

 L9 J 



L..7.-.7y:|^-^-2,,(Z,)(a''-))| 



k I 

 L V . 



We therefore write, as the covariant derivative of the system Xi of 

 order 1, the covariant system of order 2, 



Xii = -^- 2p,Xpa(p«) 



OXj 



1 3 

 L9. 



(34) 



This system may be written a little more simply by introducing the 

 Christoffel symbols of the second kind, 



2ga(P3) 



L 9 J 



_ Si 3 



(35) 



