GEOMETKIC INVESTIGATIONS ON DYNAMICS. 295 



and hence the curves .Ti are normal to all hypersurfaces .Ti = constant, 

 and they thus admit of coi normal hypersurfaces, i. e. they form a 

 normal hypercongruence ; the hypersurfaces appear as surfaces of 

 equal action. 



§7. Conditions for a normal hypercongruence. Consider 

 first a cuclidcan space of four dimensions, S4. We wish to derive the 

 conditions that a system of 00 3 curves in S4 are such that there exists 

 a system of 00^ hypersurfaces (Vs) which are orthogonal to the curves. 

 We write the equations of the 00 3 curves in the form 



(20) A' = A^ (u, V, 10, t), Y = Y (u, v, w,t), Z = Z iu, r, w, t), 



R = R {u, V, w, i) 



where u, v, iv are three arbitrary parameters whose particular values 

 Uo, vo, w'o determine uniquely a curve Co, while the variable t deter- 

 mines the points on this curve. 



We shall now assume that there is a hypersurface // which is or- 

 thogonal to the curves; let this be determined by placing 



(21) t = t (u, V, w) 



in equations (20). Through a point (A', 1^ Z, R) of such a hyper- 

 surface, determined by the values u = uo, v = Vo, w = wo, there 

 passes the curve 



(22) A' = A^ (uo, vo. Wo, i), Y = Y {uo vo, wo, t), Z = Z (uo, vo, wo, t), 



R = R (.Mo, «o, «'o, 



and the direction-cosines of its tangent are proportional to 



:^, AZ ^, ^ 



dt' dt' dt dt' 

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



/^^N dX ,^y. , dY ,-. , dZ ,„ , dR ,„ _ 



(23) — dX -\ dY -] dZ -\ dR = 0, 



^ ^ dt dt dt dt 



where dX, dY, dZ, dR are computed from (20) and t is replaced by its 

 value from (21). Now 



dX dX dX dX 



dX = — du H dv -] dw H dt, 



du dv dw dt 



