SURFACES IN HYPERSPACE. 



297 



This is a sort of partial dual of the symbol of the first kind. Then 



dX\ 



Y.. — ' _ V F 



i J} 



»j 



dX] 



'p-^i-p 



r 



(34') 



The partial derivatives dXi/dXjare expressible in terms of the derived 

 system Xij as 



dXj ( p 



(34") 



If now we take the expression VX = VyVy:(VX) + VVy(X), 

 and substitute for VVy, we have 



VX = VyVy:(VX) + | 



X X 



V 



(A-O-Vy(X) -^VyVy: 



• ( A-0 • (X) 



X X 



V 



or 



VX-i 



X X 



V 



■A-i.X = VyVy:j(VX) -f 



X X 



V 



(A-M-(X) 



)• 



If this be expanded we have, as before, 



dXi 



dxj 



<pq 



I J 



L9 J 





dXrr, 



-2 



pg 



n m 

 L Q 



(ai^P^)!, 



17. Covariant differentiation of systems of higher order. 



To find the covariant derivative of a system of the second order we 

 must substitute from (33) for the second derivatives in (31') and 

 reduce. There are two terms containing second derivatives. We 

 have 



2iy(Z.-y) -^Ts; = 2,,(X,,)T3ySp^Cp^(a(^'0 



dXrdXt 



r t 

 P . 



Ui{Xij)ysi2pqiyrpytq{a^''^)\ 



V Q 

 I 



