147 
a=a’' +a". (2) 
we have 
gs 2D ay ij = Zaria + S ae ie = Saa. (3) 
B) a e 
It is obvious that 
lenta b Litre == f 
lid = grill pige BEU yey gee Pg Bien I (4) 
and thus 
a’ a’ = a’ a’ = 0, (5) 
ap 
The quantity g’ is defined by 
2p 
gia! trapten! (6) 
P 2p 
so that by complete transvection of v with g’ we get the V,,-com- 
P 1 
ponent of v. If then V be the differential operator in the V,, V 
in the ;V„, the equation holds: 
Vp 1 Vip, (7) 
where p is an arbitrary scalar field in the V,,, and 
V'v=iV'(a!tvja=!V (a! vja = 
hi (8) 
tal nlal geur 
where v is an arbitrary vector field in the V 
Hence: 
If v is a vectorfield in Vn, V'v ts the Vyn-component of Vv. 
Pp 
The same holds, as is easy to see, for each field v in V,,. 
The curvature vector u=i! Vi of the congruence i in V,, with 
respect to Vy, or the absolute curvature vector of i can now be 
decomposed in the following way: 
wi Vi=zitvla haiti Va -.ijaS 
=i1{V@ dia t+ilfvi(a.dia’ = | (9) 
=i} Viki (Va) tia" +i (Vii a'a'= | 
== ii? (Vaja 
in which formula w’ is the relative curvature vector of i with re- 
speet to Vy. 
When we write: 
3 
H=(V'a)a’ (10) 
10* 
