694 
geodesically moved along the boundary of the surface-element do, 
is:7) 
4 
Dison = Cole el 6 5 sa 6 oo (lees) 
So (H,) demands this increment to remain in the region formed 
by i, and ij. *) 
Thus we have obtained the following theorem: 
Ill. A system of co VWV, in a Vn, whose second fundamental 
tensor, apart from determined V,,r <n, has q singly determined 
principal regions R,,,,..., Ry» but within the regions of more than 
one dimension no singly determined principal directions, belongs then 
and only then to an n-uple orthogonal system, when by moving 
perpendicular to m of the principal regions of “h, the component 
of the geodesic differential of *h, in the manifold determined by these 
m regions, has principal regions that coincide with the m mentioned 
principal regions of *h, and when besides the increment of in, when in 
is geodesically moved along the boundary of a surface-element in any 
principal region, remains entirely in this same principal region. 
14. Necessary and sufficient conditions that a V may admit n-uple 
orthogonal V,_1-systems. 
The condition (Y,") is a condition for the V, in which the z-uple 
orthogonal system exists. If we wish every system of 7 mutually 
perpendicular (n— 1)-directions in each point of the VV, to belong to 
an n-uple orthogonal V,-system, then (D,") must be valid for every 
set of four mutually perpendicular unit-vectors. It can be proved 
4 
that K can then be written in the form: 
Kele (Ff) 
in which 2? is an arbitrary tensor. For n= 3 K can always get 
this shape and, as has been proved by Corron’), every set of three 
mutually perpendicular directions in any point of a V, can belong 
to a triple orthogonal system. It can be proved that (/) is sufficient 
for n > 3 too. 
1) A. R. p. 64. 
*) An analogous geometrical interpretation can also be given to condition (D,”). 
5) E. Corron. Sur une généralisation du problème de la représentation conforme 
aux variétés à trois dimensions, Comptes Rendus 125 (97) 225—228, compare also 
E. Corroy, Annales de Toulouse 1 (99) 385—488, Chap. III. — 
