SURFACES IN HYPERSPACE. 313 



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

 1° and 2° we note that 



|.| = S„XWX(«)y,.y, = 2„X"-^X(«>a„ = S.X^'-^X, = 1, 



and similar equations hold. For 3° observe that 



t ^ 5x« ay , _ dyr 

 ds d.Vt ds 



dYr being the differential along the curves \^^\ 



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

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

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

 systems |, T|, t,, o) or |, "H, 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), 



i = 2,x(«Vrs + s.x.^y'^', 



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



^s^^'^Yrs, = S,[aX,XsX(^) + |x(X,X. + X,X.)X(«> + PXAA^^^] 



Hence 



« * ('1) 



11. = K-Xr + PX, - gipr. 



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



(55), (56), (560. 



Zi-yr = 0, Zis'Yr + Zi'Yr, = 0, 



Zis'Yr = — ibrs and Zjs'Z,- = I/y.ig. 



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

 along the normals. Hence, 



Zi^r = — ^AsY^'^ + ZjVji^rZj 



= - ^sWX^s + Mi(XrX, + XA«) 4- i3iX,X,]y(«) + 2yi/,„,z, 

 or z.i, = - i(aX + MiX.) - ^{(JiX + 0i)^r) + S,j/y,-|, Z,-. (72) 



