SURFACES IN HYPERSPACE. 307 



Thus the difference of the two third derivatives of a function is expres- 

 sible in terms of the first derivatives Xm and a combination of the 

 derivatives of the Christoffel symbols with the symbols themselves. 

 This combination is the Riemann symbol ^^ {r)ii,st\ of the second 

 kind and hence 



Xr,t - Xrts = - 2„X„{m, St] = - l\Xf»KrM, St), (51) 



where {ru, si) = Zmamu{r^n, si] (51') 



is a Riemann symbol of the first kind. As {ru, st) and {ur, st) differ 

 only in sign, we have 



Xrst - Xrts = 2uX(")(ur, st) . (51") 



From (50) we may obtain yrst — Yns and identify with 



Vrsi - Yrts = 2„y(")(wr, ^0- (52) 



As the vectors y*"^ are tangential, the components of z and w vanish 

 in this direction. Hence we obtain the equations, 



hrst — hru = CrtVs " CrsVt. (53) 



Crsl — Crts = — hrsVt " hrtV,. (53') 



{pr, st) = [(bpshrt — bptbrs) + {CpsCrt — CptCrs)]. (53") 



24. Extension to n > 4. Thus far the four dimensional case has 

 been treated. The generalization is simple. Instead of two inde- 

 pendent normals z, w, we have n — 2 normals Zi , Zo , . . . , Z„_2 and 

 may write 



Yrs = l&rsZl + 2&rsZ2 + • ■ • + n-2&«Zn-2 , (54) 



Zi'Zj- = €,;, Zi'-y, = 0, i = 1,2,. . .,71 — 2. (55) 



If we differentiate, we have 



Ziis'Zj + Zi'Z,|g = 0, Zilr'Ys + Zi'Yrs = 0, 



26 Pascal, Repertorio (Italian), Vol. II, p. 850, except for a typographical 

 error. 



