GEOMETRIC INA^ESTIGATIONS ON DYNAMICS. 297 



Since this direction is perpendicular to our surface, we must have 



(230 S^rfX< = 0. 



where dXi is computed from (20') and t is replaced by its value from 

 (21'). Now 



dXi= ''i'^dur + ^dt, {i=l,2,....,n). 



T=\dUr ot 



Introducing these values in (23'), we get 



(24') T dt -{-Uidui+U2du,+ ...■+ Un-l dUn-l= 0, 



where 



(25') T = z{'-^)\ C7,= S^4^, (r=l,2, ,.-1). 



t \ ot / i ot dUr 



The conditions of integrability of equation (24') are 



(260 T (f^ - '/A + ^^ Cf - D + ^^ if - f ■) = 0. 



\dur duk / \ dt durj \dvk dt / 



{r,k = 1, 2, , n — 1). 



' U — l\ (^ji — 2) . 

 Equations (26') give independent conditions; these 



must be satisfied if our system of curves is to form a normal hyper con- 

 gruence. 



Equations (26') are also the conditions for a normal hypercongruence 

 in any space of n dimensions, Vn, if we note that the condition (23') 

 for perpendicularity is replaced by 



(23") Saifc^rfZ, = 0, 



ik ot 



so that equations (25') are replaced by 



(25") r = S «x.^^, f/. = Sa.,^^, (r=l,2,.....r.-l). 



Xm ot ot X/i ot OUr 



§8. Analytic statement of the converse theorem. Consider an 

 arbitrary system of oo2(n-i) curves in a euclidean space of n dimensions, 



