SUKFACES IN HYPERSPACE. 305 



however, functions of xi, Xi, namely, invariant functions. The set of 

 quantities hrs , Crs are therefore covariant. As y^ = Ysr we see that 

 hrs and Crs are also symmetrical sets. 



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



Hence 



Yr'Z, = - hrs, Yr'^f, = -Crs- 



Also, from (47), 



Z'Z, = 0, wws = 0, Z'Ws+wZs=0. 

 Let 



z*w, = + Vs, W'Z, = — 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 z^ along the tangent plane and along w. Now 



y'^^-Yr = ^hyh^/^ijhir = ^htyh\ta^'"hn,\r = 2,a(p')a,, = erp. 



The solution for z^ may then be written by inspection as 



Zs= - ^pbpsY^p' - vsvr, (49) 



and checked ; in like manner, 



Ws = - SpCp^yf^') + p,z. (490 



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

 next be found by difi'erentiating (covariantly) the expressions (46). 



or 



Yrst = Z[hrst + Craft] + w[c,sJ - brsVt] " '^p[bptbrs + CptCrs]y^P\ (50) 



