686 
the two systems of surfaces, then there exists a system of surfaces ortho- 
gonal to the two given systems and the three systems intersect along 
their lines of curvature. 
For the R, this theorem has been first deduced by DARBOUX *). 
8. LinienrHaL’s conditions. We will now connect different shapes, 
in which the conditions occur in literature, for the case that i, is 
V,—1-normal, and inquire how far they remain valid, when more 
general manifolds are admitted. 
In the same way as *h the tensor *p gets a simple significance 
when i, is V;,_1-normal. Since on account of (19) and (42): 
x 
Vs 5 De VIER CN Gyn Send a ed (€) 
de 
the contravariant characteristic number of x(in. V) V ~ in is: 
DOOR Ce MGT Sia hoe! Seat vege | 
dp 
+ { (in . V) e’ en} (in 8 V) ge] + ez Ca 2 u, Un = 
=— }(in.V)* g** Heres? = (Vin)? ese (in. V)ehe' ul Epen? Urln— 
Ap 2 
= — $(in-V)? GB — 2 (Vin)? venten? (in. Vjepeng epe? UnUn = 
Ap. 
=— bin -V) GP = (Vin)? exepte’se’n® (Vin) + esex+ ep(Vin) 1 ea) | 
dp . (87) 
+ €p G24 U == 
=S Ath Weeds Sel wd on cw 2 
Mm 
+- zes! (V = in) 4 e) e’ 1 (V in) 1 Cy dE 
ar e, } (V =F in) li, in & (V in))4 €z Sin ez : (Vin)! in ip 5 (V in) 4 ex == 
= — } (in. VJ 9% + ezen? T(V ~in) 1 Vin 
from which in connection with (59) we conclude: 
~_ 
x 
p= Fe zea (in. V) gf esp (in. VAB . . (88) 
Hence the condition that *h and *p have the same principal 
directions, for the case n = 8, can be written in coordinates: 
GH gv g® 
(in. V) 9% (in. V) 9% (in. V) 9 =0, 
(in. V)? gee (in. VY)? geb (in. V)? 9% 
1) G. DArBoux, Sur les surfaces orthogonales. Annales sc. de l'Ecole Normale 
3 (66) 97—141, p. 110. 
