SURFACES IN HYPERSPACE, 305 



however, functions of .ri, x-2, namely, invariant functions. The set of 

 quantities brs, Crs are therefore covariant. As Yrs — Ysr we see that 

 brs and Cra are also symmetrical sets. 



We may differentiate (47') to find the derivatives of z and w. Then 



Yrs'Z + Yr'Zs = 0, y,-.*W + Yr'W, = 0. 



Hence 



Yr'Zs = — brs, Yr-Ws = — C„ . 



Also, from (47), 



Z'Zj = 0, wws = 0, z«Ws+W'Zs = 0. 

 Let 



Z'Ws = + vs, W'Zs = — vs- (48) 



We have then four equations (since r = 1, 2) to solve for Zs; one of 

 the equations shows that Zs is perpendicular to z and the other three 

 give the components of Zg along the tangent plane and along w. Now 



The solution for Zj may then be written by inspection as 



z, = - Sp&p,y(p^ - v,w, (49) 



and checked ; in like manner, 



w. = - SpCp.y^"^ + u,z. (49') 



23. Gauss-Codazzi relations. The third derivatives of y may 

 next be found by differentiating (covariantly) the expressions (46). 



Yrst = brstZ + CrstVr + brs^t + Crs^ft , 



or 



Yrst = Z[brst + Crsl't] + W[CrsJ — brs^t] — ^p{bptbrs + CptC„]y^P\ (50) 



