322 LIFKA. 



and on comparison with (15) we note that these are the diflFerential 

 equations of our natural family in any space of n dimensions. We 

 may therefore state the converse 



Theorem. // a system of co2(n-i) curves {one passing through each 

 point in each direction) in any curved space of n dimensions is such 

 that those oo"-i curves of the system which meet on arbitrary hyper- 

 surface {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 in a euclidean space, state a much stronger converse theorem. 



Massachusetts Institute of Technology, 

 Cambridge, Mass., January 1920. 



