597 
the vector of velocity of a streaming liquid, in which case the 
component of rotation of the movement (with respect to a geodesically 
moving system) is given by V — w‚ in this rotation every point ot 
the R, 21 V — w remains unaltered.') Indeed, dr!1(V — w) is a 
vector in the plane of V — w. 
In the same way we can prove that, if ,v is V,-creating and 
thus ,_,wW is V,-normal, the equations exist: 
| n—qW WA py =i, Qi, Sor OAT EEND (A!) 
Lv? Van vd. . Oo . . e 5 5 (B) 
in which ‚„v represents a g-vector in Va and „—,w an n—gq-vector 
LV, For p=n—1 we see from (B) that for a V,-normal vector- 
field w the component of Vw in V, is a tensor. 
3. Canonical congruences. A field of unit-vectors i, determines 
a congruence’), u,—*xi,! Vi, is the vector of curvature of the 
curves of this congruence and the modulus w, = (U,), is the geodesic 
curvature. 
As 
(V ray i, = 4 Vv (in ° i,) = 0, . 0 . : 0 . (16) 
the second ideal factor of Vi, does not contain an index n. Hence 
2 
Ji, consists of two parts, a part h in the B, Li, and a part 
16: i, V th, = zi, U: 
2 
vl,=h zi in Un, . NTS ITD (17) 
9 
In general h is the sum of a tensor *h and a bivector jh. If i 
is a unit-vector in one of the m—1 mutually perpendicular principal 
directions of *h, we have 
Se SNL rjg hol To Pt, a eee a (LS) 
and as 
VS in == th + 5 (iron + ont), . . . (19) 
we have 
ANA NA Dd a Hidde oh oe 20) 
) For Rn this is observed by A. SommerreLp. Geometrischer Beweis des 
Dupis'schen Theorems und seiner Umkehrung, Jahresberichte der Deutsch. Math. 
Ver. 6 (99) 123—198, p. 128. 
4) In A. K. p. 38 et seq. the word “Hyperkongruenz” is used. For the sake of 
simplicity we will use here the word congruence, in harmony with among others 
Moor and Levi-Civira. 
