149 
Thus the equation of an asymptotic line of, first order, 
3 LE Jg Al 
ii?H— 0 (18) 
is equivalent to the »—m equations: . 
ii? Vie= 0. | | (19) 
Since i, is perpendicular to Vz, Vi, is symmetrical in i,i,, and 
3 
hence we conclude from (17) that H is symmetrical in its first two 
ideal factors. 
When each geodesic line in Vn is also a geodesic line in V,, 
Vin is called geodesic in V,. If 0: this is Za fulfilled. 
This Binn however is also eats ise because ii H yenithics 
in this case for every choice of i, and H is symmetrical in the first 
two ideal factors. 
We thus have the theorem: 
ood Vn in the Vy is then and only then geodesic, dich H vanishes 
m sie point of the Vn. 
If H vanishes only in one point P of the Vn, the Vn is called 
geodesic in this point. This case occurs e.g. if the Vn is built up by 
geodesic lines of the V, all going through P.’) 
If in a point of the V,, the curvature vector Lied Has the same 
direction for every choice of i, the point is called awial. For mnl 
a points are axial. 
A special case occurs, aa H has the form: 
H == _ (20) 
A point, where this occurs, is called an wmbilical pomt of the 
Vn. U is called the wmbilical vector. All curves through an umbilical 
point have in this point the same forced curvature vector, and this 
vector is equal to U. } «on 
3 
For m =xn—1 H has the form: 
3 4 | 
zis EAV ii Se ct ige! bed eon <7 (21) 
The symmetrical quantity *h | Faeroe he 
4 { i 
eg Wii, vod st (22) 
!) Brancur 99, 1, p. 572 calls a Vz, in the V„ in this case already a geodesic Vo. 
