296 LIPKA. 



and similarly for dY, dZ, and dR. Introducing these values in (23), 

 we get 



(24) Tdt-\-U du -\-Vdv-}- W dto = 

 where 



(25) r = s /^T u = V ^M, V = x^^A W = S^^^\ 



\dt / dt du dt dc dt dw 



the summation extending over X, Y, Z, and R. The value of t must 

 therefore be a function of u, v, w which satisfies equation (24). In 

 order that we may have a system of hypersurfaces which are orthog- 

 onal to these curves, it is necessary and sufficient that the conditions 

 for integrability of equation (24) be satisfied for all values of t, u, v, lo. 

 These conditions are 



\dv du J \dt dv / \du 



dt / 



(26) t(^ - ^) + F (^' - ^) + W (^ - ^U 



\dw dv / \ dt dw I \dv dt I 



tI'JL _ ^JL\ +,r m _ ^J) +u /^_I _ ^) = 0, 



\du dw I \dt du) \dw dt I 



where the second and third of these conditions may be derived from 

 the first by a cyclical interchange of U, V, W and of u, v, w. 



For a cuclidean space of n divicnsions, Sn, the equations of the 

 00 "~^ curves may be written 



(20') Xi= Xi (wi, uo,. . . ., Un-h t), {i = 1,2, , n). 



The orthogonal hypersurface // is determined by placing 



(21') t = t (til, U2,. . . ., Un-l) 



in (20'), and through a point Xj^"^ of such a hypersurface, there passes 

 the curve 



(22') Xi= Xi Cui(«), W2W,. . . ., w„_l^«^ 0, a = 1, 2,. . . .n), 



and the direction-cosines of its tangent are proportional to 



^'{i-l,2,....,n). 

 dt 



