SURFACES IN HYPERSPACE. 313 



to the curves X^'^^ and to their orthogonal trajectories X'*"). To prove 

 1° and 2° we note that 



|.| = 2„X('-)X<»Vr-ya = 2„X('^X(«'a,3 = ^r'K^'K = 1, 

 and simihir e(|uation.s hoUl. For 3° observe that 



ds dxt ds 



dyr being the diti'erential ah)ng the curves X^''^ 



In case of four dimensions we shall use '^, w (to correspond with |, 

 11) in place of z, w as the unit normal vectors — in higher dimensions 

 Zi, Zo, . . .z„_9. We have therefore such relations as (47) or (55). The 

 systems |, r\, t,, o> or |, t], z,, i = 1,2,..., n—2, are therefore systems 

 of moving axes in which |, t] move along definite orthogonal trajec- 

 tories upon the surface. 



The rate of change of the unit vectors |, t\ are, by covariant differ- 

 entiation of (70), 



t = 2,X(«Vr3 + ^.x„y(^', . 

 From (68), (61'), (62), 



2«X(«)y„, = 2, [aX.XsV^^ + \i{K\s + XrX.)X(*^ + PX.X,X(»)] 



= O-Xr + \t'\r = 2i(aiZiXr + jJLiZiXr), 



Hence 



t - aX, + |i.X, + tl^„ 



ft— *. (''1) 



11, = ]IX + PX, - i<Pr. 



The rates of change of the normals are found from the relations 



(55), (56), (56'). 



Z.-y,- = 0, Z,-3*yr + Zi'Yr, = 0, 



Zis'Yr = — ibrs and Zi^-Zj = Vjii,. 



These equations give the components of Zi^s along the surface and 

 along the normals. Hence, 



Zt|r = — S^^r^y*'^ + SyZ^jiirZy 



= - 2s[aiXrXs + IJii(Kr\ + XrXs) + /3tX,X«]y(*> 4- ^jVjtir Z, 

 or Zur = - tiuiXr + MtXr) - T]( ^ii\r + /3iX,) + ^ jV ii\r Zj. (72) 



