602 
HOW ih) eS Ee Cis a ae (HN) 
This equation however is of the shape (20) and hence each of the 
desired vectors sz forms one of the canonical congruences belonging 
to i,. At first we consider the case that the n—1 roots of (24) are 
all different. In this case every vector 8; must be equally directed 
with a definite i: 
1 
ig = OkSk=— Si? - ol or (62) 
Ok 
The n—1 canonical congruences belonging to i, must therefore 
all be V,—1-normal. In order that this may be so, i, has to satisfy 
certain conditions that may be obtained as follows. 
Application of V to (31) gives: 
(Tij)! (Vin)! ie + (Vint (V ~ in) 1 ij {VT il? ir = 09%, (53) 
and transvection with in: 
int (Vit (V in)? ie + int (Vip) (V — in)t ij Hij ir in? V(Vin)=0.(64) 
In consequence of (46) and (51) (V~i,) 1 iz contains only iz and 
i,. Further i; is V,-1-normal, so that according to (B): 
int (V ij)? ik =iz 1 (V i;)! bon Se (55) 
Hence (54) is equivalent to: 
— iz! (V ~— in)! (Vi)! ij—ij} (Vi)! (Vi)! iz aa i; ik i, 3 V7(V~i,)=0. (56) 
If now 
1,...,n—1 
tn an = Dn Dn == = ij ij eere c z 5 (57) 
J 
we have a quantity: 
4 
Sn — an Dr b, An. : . . . . : : : . (58) 
which when transvected twice with an arbitrary affinor of second 
degree gives the component of this affinor in the A, 1 Li, Intro- 
ducing the tensor: 
4 
De WLA SE oe GD) 
we get from (56): 
iin? *p = 0, Wc le li oe atthe Ca) 
Hence the first condition is that the tensor *p has the same principal 
directions as *h. 
Since on account of (19): 
4 4 
En 2 (in . V) (U xe in) = Sn 2 (in . V) *h AF Xl Uns. 0 5 (60) 
und on account of (19) and (30): 
