316 LIPKA. 



Hence, the 0's are partial derivatives of a single function L{xi,X2, . . ., 

 Xn), or 



d T 



(85) <l>i= —, {1= l,2,...,n). 



dxi 



Introducing these values into (82), we finally get 



(86) x"i= ^^ -x'ii: ^^xi {i=l,2,..., n), 



OXi k OXk 



and on comparison with (13), we note that these are the differential 

 equations of our natural family in a euclidean space of n dimensions. 

 We may therefore state the converse 



Theorem. // a system of 002(^^-1) curves {one passing through each 

 point in each direction) in a euclidean space of n dimensions is such that 

 those cx>"-i curves of the system which meet an arbitrary hypersurface 

 {space of n — 1 dimensions) orthogonally, always form a normal hyper- 

 congruence, the system is of the natural type. 



The Lipschitz theorem (§6) shows that the systems of the natural 

 type actually have this property. Since we have only considered the 

 vanishing of ai in the condition (38), we may here, analogous to the 

 case of four dimensions, state a much stronger converse theorem. 



§11. Proof of the converse theorem in any space. In §7 we 

 derived the necessary analytic conditions 



(r = 1,2,..., 71- 1), 

 for a hypercongruence of curves 



(200 Xi = Xi {uu U2, ...., Un-i, t), (i=l,2, ....,n) 



to be of the normal type. We may proceed exactly as in §8 and apply 

 these conditions to a family of 00 "-1 curves picked out from the system 

 of oo2("-i) curves defined by 



