604 
The two conditions (C) and (D) are not only necessary, but also 
sufficient. Indeed, from (C) and (C”), which are perfectly equivalent, 
we conclude (56) and from (30) we conclude (54). Comparing (54) 
and (56), we get (55). From (D) we get, when comparing with (65), 
which results from (30), the equation (67). But (55) and (67) show 
that congruence i; is V,,y-normal. 
Finally we have the result: 
The necessary and sufficient conditions that we can bring n A 
mutually orthogonal V,—, through the congruence i,, whose corres- 
ponding algebraic characteristic equation has but unequal roots, are 
that in satisfies the equations (C) and (D) 5. 
(n—-1)(n —2) 
9 , the number of 
The number of the equations (C’) is 
(n—1)(n—2)(n—38) 
2 
j and & may be interchanged without creating a new equation, but 
not j and /*). Considered as differential equations in the characteristic 
numbers of i, both systems (C) and (D) are of second order. 
the equations (D) is , according to the fact that 
6. Simplifications for the case that the given congruence is Var 
normal. If i, is V,—1-normal, then ,h —0, and ’g, is the first and *h the 
second fundamental tensor of the V,,-1; Lin. Its principal directions 
determine the directions of principal curvature. (61) changes into: 
4 
$n AT (Y = in) (Vin) 27h? Huan - (69) 
and (62) into: 
4 
*D = gn? (in . V) *h — 27h! *h AT NT (70) 
(C) changes into: nae 
ij iu 2 (in . V) MW 0, 5 5 5 5 e 5 5 (C,) 
and gets with this the same shape as (JD). 
In the same way as with (68) we see here: 
Beane) Ws (no VANGEN 5 vee 16 (Tl!) 
hence also the principal directions of (i, . V) 7h are singly determined. 
1) (C’) is deduced for the first time by Rriccr, Dei sistemi, vergel. (A), p. 309. (D) 
has in his paper a less simple shape, in our notation: 
ij iz? V (V — ir) = hie um) ij iu? Vin + 4 G+ ui? Vin, (DD) 
and is denoted equation (B), p. 309. (D’) results when we apply the operation 
(i. V) to (31). (C) and (D) are consequences of (30), (C’) and (D’) of (31). Here 
we have first deduced (C’), because the condition in this shape is identical 
with the condition given for Rs by LittenrHaAL, which is very important for the 
problem, as may be seen in the second part of this paper. 
*) G. Rreer, Sui sistemi, p. 152. 
