SURFACES IN HYPERSPACE. 



299 



In this particular case the system is symmetric, A'„ = Xsr, because the 

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

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

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

 derivatives of the same function F. 



18. Contravariant differentiation. If A"''^) is a contravuriant 

 svstem of order 1, \ve should call a contravariant set X'"' of order 2. 

 which contains the derivatives dX'-^^/dXs and the coefficients a^ and 

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

 this set by considering the dual X'^"'> of the first covariant derivative 

 Xrs of the set Xr , dual to the given set A"^''^ Thus, 



ZC"") = i:rsa^"'^a^''^Xrs = 2„a('-")a'"') 



dXr 



dXs 



— 2 Y 



r s 



m 



Xr — a^^^'^ZmXr.i ■) \ 



dXs dXs ( m ) 



{rsl 



Z,a^''-^ 



— ZrtX^'^Ort— 2„„,a('-")Z(')o,„,] 



dxs OXs ( m 



Now as Srarffl^''"^ = ^ut , we have 



aa^"-") ^ , .dart 



rd/t' 



dx.. 



Hence 



dXa 



- I,.,a('-")X") 



r s 

 t 



Hence 



X(uv) = XM'"^ 



az(") 



a.r. 



+ 2 





(32') 



(37) 



21 See Ricci, Lezioni, p. 70. 



