598 
or 
(C2 7th A ETE 6 0 bo 2 oo (2115) 
for 
TEE Te (DE) 
From (21) follows: 
in! (xV = in — 2 1) =e Wi helen Saad & (23) 
or, when X, are the covariantive coordinates of V ~i, and 7, 
those of in, in coordinates: 
0 Ühahosd 6.6 oo Dore 0 tna, 
Un a Xa, En A Yara, « Xa, ae Ja, a, 
—= 0. (24) 
a X == eke qe —A 
nay, ony Jae, tn Ja, a, 
This equation of degree n—1 in / is called by Rrcor the algebraic 
characteristic equation of the congruence i,. *) Since the tensor *h has, 
as is known, n—1 real principal directions, the equation (24) has 
n—1 real roots.?) When all roots are distinct (that is when no two 
roots are equal in all points of V,, which does not exclude that they 
may be equal in some manifolds of less than n dimensions), we see 
from (21) that to a definite root 4; belongs the direction: 
SOC typ = data" ies 0 ve ove (8) 
Two directions belonging to distinct roots are mutual perpendicular, 
because with regard to (20): 
Zins We == ais (WY LE es Aptis a, HA so (PP) 
A p-fold root determines a region fi, perfectly perpendicular to 
the regions of the other roots, and in this region we may choose 
p arbitrary mutually perpendicular directions as principal directions. 
In every case we can indicate to the given direction i, in every 
point n—1 mutually perpendicular principal directions that join to 
the congruence i, n—1 mutually perpendicular congruences ij, j=1,2, 
...,m—I1. Riccr calls these congruences the orthogonal canonical 
congruences belonging to in. *) 
1) G. Ricci. Dei sistemi di congruenze ortogonali in una varieta qualunque, 
Memorie R. Acc. Lincei Ser. V 2 (95) 276—322, p. 301. 
2) For a direct proof see e.g. G. Ricci, Sui sistemi di integrali independenti di 
una equatione lineare ed omogenea a derivate parziali di 1° ordine, Ann. di Mat. 
Ser. II 15 (87/g.) 127—159, p. 134. 
3) G. Rreer. Dei sistemi, p. 302. For the sake of brevity we will speak here of 
canonical congruences. See also G. Riccr and T. Levi Crvira, Méthodes de calcul 
differéntiel absolu, Math. Ann. 54 (Ol) 125—201; J. E. Wricut, Invariants of 
quadratic differential forms. Cambridge Tracts N°. 9 (08), p. 73. 
