ESO Oy 2 8 Or OLE Win Gro 2 Vinh = 
Ì 
S=— OF Ge PW de EV = 
Zen 2e, ig 
1 (94a Òaa 1 ade da; . . (48) 
n dE n i aie A 
Ss || Se a) an |= a) A 
Qe, \ Oat Ow) Zen \Qatn Aan 
a : 
=—(e'o . V) 0p =FH(in - V) Giz. 
DE ow 
Hence *h is the second fundamental tensor of the Va Lin’). 
When i, is geodesic without being V,,-;-normal, then we have 
u, =O and Vi, lies totally in the region 1i,: 
wi Baalen 70 (4) 
When i, is normal too, Vi, is symmetrical : 
VAi Ne ae ee (4D) 
In this latter ease choosing 2°" as the length measured from a 
definite WV, Li, along the curves of the congruence i,, we get: 
ea, = Ea, Sin - - - . « « « (46) 
5. Mutually orthogonal V,y-systems through a given congruence 
when the canonical congruences are singly determined. 
When given an i,, it is required to choose the original variables 
y',..-y"—1 in such a way that the corresponding aequiscalar Va 
pass through 1, and that the vectors s=Vy,j=1,..., n—-1, 
are mutually perpendicular. 
Hence the system of equations: 
heal == oar spe a aT) 
Citic SY (iO aes HP San) ae Soa ine wena) — alt) 
must allow for every value of 7 m —2 independent solutions. The 
necessary and sufficient condition is, according to (7): 
in) Vers! Vin=arsr Honing - . » « (49) 
in which «, and «, are arbitrary coefficients. Since in LS, and 
accordingly : 
in) V sp =(V sz)! in = V (Sp. in) -— (Vin)! sr = — (Vi)! sz, . (50) 
(49) is equivalent to: 
') Compare Biancai-Lukat, Vorlesungen über Differentialgeometrie (1899) p. 
601, form. (7). The principal directions of curvature may also be defined as the 
principal directions of the second fundamental tensor. So Brancut, p. 609, 618. 
