SURFACES IN HYPERSPACE. 



297 



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



A^,= ^-2,A% \'H. (340 



d.Vj ( p ) 



The partial derivatives dXi/dxj are expressible in terms of the derived 

 system A'i,- as 



dxj ( p 



(34") 



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

 and substitute for Wy, we have 



VX = VyVy:(VX) + i 



X X 



V 



(A-i)-Vy(X) - §vyvy: 



'(A-i).(X) 



X X 



V 



or 



VX-i 



X X 



V 



•A-i-X- VyVy:J(VX) -^ 



X X 



V 



(A-0 . (X) 



If this be expanded we have, as before, 



dXi 



dxi 



'PQ 



L9 J 



i ri V 



a(3P)A^p = Y,„,njimyin -j r^" 



f OXm 



'PQ 



n m 

 L 9 



{a^<'P^)X, 



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 



iiiXid ^y.i = ^i,{X,;)y^Z,ic,M''') 



dxrdxt 



r t 

 IP ] 



- ^ai^ iihsH pqarpi ta{a^'^^) I 



P Q 

 I 



