SURFACES IN HYPERSPACE. 291 



Xi, = dXj/dXj, would form a system of the second order. Let us 

 consider the transformation of this system. 



Xi, = -— -t7.A- {Xk) = ^klik — h -A: (Afc) -— 



dXj dXj dXj dXj 



^ d(Xk) di/i djik 



= ^kjik^i ^ — -- + -t (A A-) — - 



oi/i dXj dXj 



= 27,/:7// (X.,) + 2. (A'.) ff- . (31) 



If it were not for the second term, the transformation would be co- 

 variant, but the presence of this term shows that the derivatives of a 

 covariant system of the first order do not form a covariant system of 

 the second order. 



The same is true for covariant systems of any order, — their deri- 

 vatives do not form a covariant system. For instance in the case of a 

 covariant system of the second order Xrs, by a similar transformation, 



d^yi dyj dhjj diji' 



— Zijkyri^srYtk — ;, -T ity(A ij) 



oxt dijk 



where dyi/dXg = ysi and d'^yi/dxrdxt = djri/dxt. 



dXrdxt dXs dXsdXt dXr 



, (31') 



The fundamental relation dxi = Hidjdyj may be written in matrical 

 notation as rfx = c/yVj,x. It follows that Mc = Vj/X. We may also 

 write dy = f/x-VxY- Hence VxY and Vj,x are reciprocals. The rela- 

 tions (29) and (30) may be written as 



X = Vxy(X), XY = V.yV.y:(XYj, 

 (X) = V,x.X, (XY) = V,,xV,x:XY, 

 X° = (X°).V,x, X°Y° = (X°Y°):V,xV,x, 

 (X°) = X°.V.y, (X°Y°) = X°Y°:VxyVxy, 



and so for systems of any order. 



The differentiation of a system of the zeroth order /is accomplished 

 as: 



df=d{f), dx'Vj^dyV.if), 



rfx-Vj = dx-V^yVyif), Vf = Vxy(V/). 



