148 
we have: 
3 
pee en! (11) 
t a 3 
In this equation wu’ —ii*®His a vector perpendicular to V,, a’ 
containing but in4:1,....,in. We call u’ the forced curvature vector 
(erzwungener Krümmungsvektor) with respect to the V, and H the 
curvature affinor of the V,, with respect to the V,,'). We thus have 
the theorem : | | 
‘The absolute curvature vector of a curve in the V,, 1s the sum of 
the relative curvature vector*) and the forced. curvature vector 
with respect to the Vy. 
‘u’ vanishes for a curve, geodesic in V,,, thus u’ is the absolute 
curvature vector of a curve, geodesic in V,,, with the same tangent 
vector as the given curve. If u” vanishes, the curve is called 
asymptotic line of first order of the V,,. Hence a geodesic line in 
V,, is then and only then geodesic in JV, when it isan asymptotic 
line of first order of the Vn. 
3 
§ 3. Theorems concerning the quantity H. 
Because of 
(Va)a=0') (12) 
we also have 
(yt) ae 0 beh 69) 
; 
and thus we can give to H the following form: 
33 4 4 4 
H — g'2 (V a) a’ = EN s 2 (V a") a’ — __ g2 V a’a". (14) 
Since however : 
; TRE EN eT) SES GTi ee SN (15) 
3 
we can get another form for H: 
ee 
Hg VE. (16) 
From (14) still another form can be deduced : 
3 4 4 4 
H=—¢?V Ziip = Egt Vie Eg (Ti)ie. (17) 
3 
1) H corresponds with the expression (2,1; in Voss, 80, 1, with bajrs in Rrccr 
03, 2 and with KY) in KüHne, 04, 1. 
3) Riccr, 02,1 calls this vector “‘curvatura normale relativa a Vp”. 
La) Oe p- Za 
