298 LIPKA. 



assuming that one curve passes through each point in each direction. 

 Such a system may l)e defined by a set of n differential equations of 

 the second order — 



(27) x"i = Fi (Xi, Xo, ,Xn] x[, X2, Xn), {l = 1,2, , Tl), 



where x'i = -^,x"i = — —, and the arc length s is chosen as parameter 

 (Is f/^- 



along the curves, so that 



(28) S (.rO^- = 1. 



i 



Here the Fi are uniform functions, analytic in the 2n arguments. 

 Our problem is to find the form of the function Fi so that the oo"-i curves 

 of the system ivhich pass out orthocjonaUy to any hypcrsurface in the space, 

 should form a normal hy per congruence. 



Denoting the initial values of Xi, x'i, which may be taken at random, 

 subject to (28), by Xi, pi respectively, and employing A'l, X^,. . . ., X„ 

 as current coordinates, we may write the solution of (27) in the form 



(29) Xi = Xi+ Pi is -s)-\-hFi{S- sY + i Mi (S -s)^ +...., 



(i= l,2,...,n), 



(30) 'Xpi^=l. 



i 



Here the Fi are expressed as functions of Xi, pi, and the Mi found by 

 differentiating (27) are given by 



(31) Mi=x(^/^p,-\-^/^F), (i=l,2,...,n). 



I \dxi dpi I 



Consider, now, an arbitrary hypersurface 



(32) Xi= fi (ui, ui, , w„_i), (i = 1,2,..., n). 



At each point of the surface and normal to it a definite curve of the 

 given system (27) may be constructed. A certain hypercongruence 

 will thus be determined. We wish to express the conditions that this 

 shall be of the normal type. If the direction of the normal at any 

 point of the surface is given by Pi (ui, U2,..., Wn-i), {i = 1,2,...., n), 

 then these Pi are determined by the equations 



(33) - SP.^=1; SP,§^ = 0, ir=l,2,....,n-l). 



i i aur 



