GEOMETRIC INVESTIGATIONS ON DYNAMICS. 299 



The second of these conditions merely expresses the fact that the 

 normal is perpendicular to the parameter curves of the surface (32). 

 By differentiating (33) we have the further relations 



(34) 2 Pi- — = 0, (r = 1, 2,. . .,n — 1); i- — — - = S 



i dUr i Olllc OUt i OUr OUk- 



(r,k=l,2,...,n-l). 



We may now identify the normal at any point of the surface with the 

 tangent line to the curve, so that 



(35) Pi = Pi (mi, U2, , Un-\), (i = 1, 2, ,n). • 



The equations of the oo"-i curves corresponding to the given initial 

 conditions may now be written 



(36) Xi= fi+ Pit ^ ^Fif-h iMif-\- . . . ., (i= 1,2,. ..,n), 

 where t replaces S — s, and where, e. g., 



(37) Fi{uuU2,...,Un-l) = Fi{fi,f2,...,fn; Pi, P2,....,P„). 



The coefficients of the powers of t in (36) are thus functions of (n — 1) 

 parameters Uu tio,- ■ ■ ■ , Wn-i, so that the equations of the oo"-i curves 

 are expressed in the form (20')- In order that the system (36) form a 

 normal hypercongruence, equations (26') must be satisfied. 



Applying (25') and (26') to (36), and using the conditions (33) and 

 (34), we get 



dt dUr dur dUr dUk dUk 



+<p+.... 



OUk 



r = 2 P^+ 2t'2PiF^-h .... = l-\-2ti:PiFi-{- ...., 



i i i 



11 r = 2 P,^ + t ^2 P,^ + 2 Fi^-f) +....=/ 2 P.^ + . . . ., 



i OUr \ i OUt i OUr/ i OUr 



C7, = 2 Pi^-^+t (z Pi^ + 2 Pi^V- • • • =^S P,^ + . . . 



i OUk \ i OUk i OUk/ i OUk 



f =2, (z P..|t +2 F.p) +....:'-L =2, (s p/f + ^-F.p) 



OUr \i OUr »' OUr/ OUk \i OUk i OUk/ 



+ .... 



