600 
rotation of i, when moved in the f#-direction occurs in the nk-plane. 
When we define the principal directions of curvature in a point 
of the V,-; as the directions i, in which the geodesic rotation of 
i, occurs in the nk-plane’), we can conclude that the canonical 
congruences are the principal directions of curvature of the V‚ a LAn. 
Still a consequence of (29) and (83) is 
zij! V in = 7; ij, Hells 9 dlg NSS) 
equation equivalent to (37), which may also be considered as definit- 
ing equation of the principal directions of curvature of the V,,_; 1 i,. 
4. The second fundamental tensor of the V,—, Lin. In order to 
make clearer the signification of *h, we choose the V, 4 Li, as 
aequiscalar regions of the original variable 2’n and n—1 arbitrary 
systems of VV, through i, as aequiscalar regions of the original 
variables a’, 2 = a,,...@,—1. Then the directions of e, and e,’ lay 
in the Vl i,, while @, and ea have the direction of i,: 
. a ij 
Sh SG VY G8 Ön eg 
n En n 
LOE 5) 
Then the contravariant 2u-characteristie number of 7h is: 
TPP == HGn? SHOES Wid Fr With 6 va (0) 
or, because i, Le and | es, and: 
VASES) D=Ggooso Gein o oo (EN) 
also: 
"Y= — tinen* Ver tire? Ver 
1 
=— (ese 2 Vetere? Ve) = 
Ze we 
1 da dat ) cS 
== ae + Coal 
Zen ‘= 0 fou 
In the same way the covariant characteristic number of *h is: 
1) This definition is the natural extension of the definition of the lines of 
curvature on a Vs, in Rg as the lines, along which the normals form a developable 
surface. 
