691 
Since: 
V ' (V in) a (Von | — (VV in)! WV On + a(V in) 1 (a 5 V) V On, (112) 
we have: 
4 
Vl(Vin) 1 V oni =BY | (Vin)! Von} = vis K2in,Vor—B (Vin)! VV on,(113) 
so that (111) is equivalent to: 
4 14 4 
gs? |- Vv (Vin) 1 V on} + 2 K2 in V On—6, B(V in) in? K! is [=0. (114) 
Since in a space of constant Rimmany-curvature on account of (92) 
and (101): 
Ay 8 4 4 4 
g? K? In WV On = — On £2 K? in Un = — 20, K, 8? in — Ur = 0, (115) 
the condition for such a manifold is, on account of (110), that the 
component of V — {(Vin)! Von} in the region Jin vanishes. On 
account however of Strokes’ law‘), we have for each vector v: 
fra zaf tt Oo 0d zee (1811) 
s 5 
in which s is a closed curve and *fdo the bivector of the surface- 
element of any surface o bounded by this curve. From this we 
conclude that in a space of constant Rimmann-curvature we can also 
give as first condition that the linear integral of the vector (Vin)! Von 
along each closed curve in a Va Lin vanishes. This condition is the 
only one for V,. Foran R, it has been first indicated by WeEINGARTEN *) 
and Ricci *) has observed on occasion of WeINGARTEN’s paper that the 
condition holds also for a V, of constant Rimmany-curvature. From 
the above-mentioned we see that the condition, but no more as the 
only one, holds also for manifolds of constant Rimmann-curvature, 
for which n > 3. 
1) Comp. A. R., page 37 and 61. 
2) Weincarten, Ueber die Bedingung, unter welcher eine Wlächenfamilie einem 
orthogonalen Flächensystem angehört. Crelle 83 (77), 1—12. 
*) G. Rreer, Della equazione di condizione dei parametri dei sistemi di superficie, 
che appartengono ad un sistema triplo orlogonale. Rendiconti Acc. Lincei Ser. V, 
Ill, (94) 93—96. 
Ricct observes for the case n=3 that Wrincarren’s theorem remains also 
4 
valid, when K has the shape: 
4 
== i (a — b) (a ~ b) + » (1, 1,) (1, SH) 
when v is an arbitrary coefficient. This however holds also for general values of ». 
