605 
In connection with the already given geometrical interpretation 
of (D) we get the following theorem: 
I. A system of @' V,-1 in a Vy, whose second fundamental 
tensor *h has n—1 principal directions that are singly determined except 
on determined V,,r << n—1, belongs then and only then to an n-tuple 
orthogonal system, if the component of the geodesic differential of *h, 
when moved perpendicular to m of the principal directions of *h, 
has in the R„ determined by these m directions principal directions 
coinciding with the denoted m principal directions of *h. 
This theorem is given for a system of V, in R, by Maurice 
Levy °). 
When the principal directions of *h for a certain point P are 
not singly determined, we conclude from (68) that also the principal 
directions of x (i,.Y)*h, and hence those of *h, for all points of a 
curve of the congruence i, through P are undetermined in the 
same way. From this we see: 
II. In a system of w* Va in a V, belonging to an n-tuple 
orthogonal system all points in which all or some directions of 
principal curvature are not singly determined (umbilics in a wider 
sense) are arranged on loci consisting of curves of the congruence 
orthogonal on the Vs. 
Also this theorem is first given for a system of V, in R, by 
M. Lévy. ”) 
Since the characteristic numbers of i, may be expressed in the 
first differential quotients of the parameter determining the system 
of the V,—;, the equations (C) and (D) are partial differential 
equations of the third order in this parameter. *) 
hj M. Lévy. Mémoire sur les coördonnées curvilignes orthogonales. Journal de 
Ecole Imp. Polytechnique 26 (70) 157—200, p. 159. 
*) M. Lévy. Mémoire etc. p. 174. 
5) For a short survey and a discussion of the literature of this differential 
equation of the third order for a system of Vs in Ry [for here (D) disappears and 
the system (C) reduces to one equation] see e.g. Lucien Lúvy, Sur les systèmes 
de surfaces triplement orthogonaux. Mém. couronnés et Mém. des savants étrangers, 
Bruxelles 54 (96), 89 p., p. 5 and following pages. 
40 
Proceedings Royal Acad, Amsterdam. Vol. XXII. 
