308 LIPKA. 



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

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

 We may therefore state the converse 



Theorem. // a system of co^ curves {one passing through each point 

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

 those 00 3 curves of the system which meet an arbitrary hypersurface 

 (space of three 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. In our discussion we have only 

 considered the vanishing of ai in the condition (38) for a normal 

 hypercongruence. We may therefore state a much stronger converse. 



Theorem. If a system of co^ curves in a euclidean space of four 

 dimensions is such that those oo^ curves of the system which meet an 

 arbitrary hypersurface orthogonally, always meet some infimtcsimally 

 neighboring hypersurface orthogonally, the system is of the natural type. 



As stated in the last theorem the weaker requirement of orthogonality 

 must hold for all hypercongruences of the system, for a hypercongru- 

 ence may meet two neighboring surfaces orthogonally (i. e. be approxi- 

 mately normal) without meeting «3^ hypersurfaces orthogonally. 



§10. Proof of the converse theorem in a euclidean space of n 

 dimensions. We return to our (?i — 1) in — 2)/2 conditions 



(41) s^f^^-^^W2^(^^^-^^Vo 



ildXi \dUrdUk dUkdUrJ ildpi \dUrdUk dUkdUrJ 



(r,k= 1,2, ...,n-l) 



for a normal hypercongruence. We may evidently place each of the 



two summations in the left member of this equation equal to zero. 



By grouping the terms in the first summation, this may be written 



ii \o.Xi axij \dUkdUr aUrOUkJ 

 To eliminate the/'s we proceed as follows. Solve the {n — 1) equations 



(33) SPi^=0, (r = l,2,...,n-l) 



t OUr 



for the ratios of the P's. If we use the notation 



