290 



WILSON AND MOORE. 



16. Covariant differentiation of a simple system. Let us 



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



^^' = ^kaikln ^^ + ^,{Xu)^aCtM''') 



dxi 



dyi 



— Si.(Zi.)SrsZ7trTys(« 



t 



{km 



I 



or 



dXj 



dxj 



— ^ki^kj'^tlCtiy 



a-.) [y] 



= ^klJikJil 



d(X,) 



- ^t{Xt)^kipyikiM"'^) 



k I 



I p ^ 



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 (Xk) by its value ZmXmCmk and (a^^^^) by its value 

 2pga^p^^7pfc73Z, we find that 



k I 

 . p. 



We therefore write, as the covariant derivative of the system X< of 

 order 1, the covariant system of order 2, 



OXj 



.9 J 



(34) 



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

 Christoffel symbols of the second kind, 



Sgtt^PS) 



^ J 



Ip 



(35) 



