690 
4 
1j in? VV On = 2 “On® ij iz ? {sn 1 (V ae V) Sn} = KOn ij in? (in 1 Ki in) (109) 
or 
4 
i; ix 2 VV on=xon in i; iz in 4K. 
(C,) 
Thus for a V,, for which the characteristic numbers (nkjn) of 
4 
K vanish, this first condition can be written: 
li; ie Oo WW Op == 0. | (C",) 
This equation expresses that the tensor VVon has the same prin- 
cipal directions as *h. The geometrical signification of o, is that 
this quantity is proportional to the infinitesimal distance between 
succeeding V1 Lin measured along in. 
In space of constant RimMany-curvature AK, we have, in connection 
with (101): 
ii? fin! kt in} = — Ko ij de? (°8 — in in) —0, . . (110) 
from which we conclude that in this manifold the first condition 
has the shape (C,), hence also in euclidean space. In this 
latter case the condition is deduced for n=8 by Levy’), 
CayLey’), DARBOUX®), and for general values of n by Darpovx f). 
Thus the necessary and sufficient conditions for manifolds of constant 
RreMANN-curvature are (C,') and (D,’). 
11. Wertearten’s condition. We will try to find a shape of the 
conditions that only depends oni, and no more on ij, j=1,2,.. ‚nd. 
When a tensor, whose principal directions do not coincide with 
those of *h, be transvected once with *h, an affinor arises whose 
alternating part is certainly not annihilated. Thus the condition that 
the principal directions coincide, is that the alternating part of the 
first transvection with *h vanishes. Hence (109) is equivalent to: 
4 4 
Bg? {(V in)! (VV on) — on (V in) In 7K inf} =0, . . (111) 
in which B may indicate that the bivector-part has to be taken. 
1) M. Levy, Mémoire etc., p. 170. 
2) A. CAYLEY, Sur la condition pour qu'une famille de surfaces fasse partie d'un 
système orthogonal, Comptes Rendus 75 (72), a series of articles. 
5) G. Darsoux, Sur l'équation du troisième ordre dont dépend le problème des 
surfaces orthogonales. Comptes Rendus 76 (73) 41—45, 83—86. See also e.g. 
Brancur-Luxatr 1st. german edition. 
4) G. Darpoux, Legons sur les systèmes orthogonaux etc. p. 128. His formula 
(32) is our formula (C’,). 
