SURFACES IN HYPERSPACE. 



299 



In this particular case the system is symmetric, Xrs = A'g,-, because the 

 Christoffel symbols are, as is known, symmetric. Moreover ^^ if 

 the covariant derivatives Xrs of a system Xr form a symmetric system 

 Xt8 = Xsr, then the elements Xr of the system, must be the partial 

 derivatives of the same function F. 



18. Contravariant differentiation. If X'-''^ is a contravariant 

 system of order 1, we should call a contravariant set A'*"' of order 2. 

 which contains the derivatives dX^'''>/d.Vs and the coefficients o„ and 

 their derivatives, the first contravariant derived set. We may obtain 

 this set by considering the dual A"'") of the first covariant derivative 

 Xrs of the set Xr, dual to the given set X^^\ Thus, 





dXr _ ^ ir s 



dXs { m 



dXs o.Vs ( m 



= 2.sa(«''^ 



dXs oxs ( m 



Now as 'Erttrta^''"^ = eut , we have 



rttrt 



dx, 



= - 2,a(''") 





Hence 



Xiuv) ^ 2.,a(^''l 



^.Tg dXs 



r s 

 t 



But, by (32), 



Hence 



dart 

 dXs 



Xt"") = S.a^*") 



r .s' t s 



t \ ~ I r 



(32') 



(37) 



21 See Ricci, Lezioni, p. 70. 



