SURFACES IN HYPERSPACE. 307 



Thus the difYerence of the two third derivatives of a funetion is expres- 

 sible in terms of the first derivati\'es A',„ and a combination of the 

 derivatives of the Christoffel symbols with the symbols themselves. 

 This combination is the Riemann symbol ^^ {nii,st} of the second 

 kind and hence 



Xr^t - Xrts = - ^,nX,n{nn, ^i] = - ^uX^'^KrU, d) , (51) 



where {ru, st) = 1,„,amu{n», st} (51') 



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

 only in sign, we have 



Xrst - A%,, = 2„Z(")(Mr, st). (51") 



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



Yrst - Yrts = 2„y(«)(ur, st). (52) 



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

 in this direction. Hence we obtain the equations, 



brst — brts = CrtVs — CrsVt- (53) 



Crat — Crts = — brs^t — hrtVf (53') 



{pr, st) = [{hpjjrt — hpthrs) + {CpsCrt — CptCra)]- (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 



yra = \hrsZ^ + ihrs^'l + . . . + n-lhrs^n-2 , (54) 



Zi'Zj=€ij, Zi'Ys = 0, i = 1,2,. . .,n — 2. (55) 



If we differentiate, we have 



Zils'Zj + Zi'Z;|s = 0, Zilr'Ys + Zi'Yrs = 0, 



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

 error. 



