160 
For ‚a: Vest jin the Ve Gue passes for a even into o, and a, Oge 
passes for a + 8 even into o,0s. Hence in this case all quantities 
6. for n >> 3 and all quantities o., a >1 for n = 3 are only depend- 
Bla 
ent on the linear element of the VV, and on g’4K. This theorem 
can be proved in a simpler way for n>3. After (39) *h is 
t+ 8 4 
the vector-tensor belonging to the bivector-tensor 4 (K’—g’ * K.) 
Now a vector-tensor is then and only then uniquely determined 
by its corresponding bivector-tensor, when its degree of nullity is 
n, n—1 or n—2. For n > 37h is thus in general uniquely deter- 
4 8 4 ks 
mined by K’—g’ *K and consequently also all principal curva- 
tures and all quantities o, From this follows that a V,—, in 
8 
R, or S,, in which Kg ‘K only depends on the (n—1)-direction 
of the V,-,; in P, cannot in general be deformed’). We can 
observe that for the same reason a V,, in Rn or S, with only 
axial points cannot in general be deformed in such way that its 
points remain axial. In a deformation of a V,, in V, also an 
axial point remains in general not axial. 
LITERATURE. 
1869. 1. KRONECKER, L. Ueber Systeme von Functionen mehrerer Variablen. 
Monatsber. Berl. Ac. 159—193, 688—698, Werke I, p. 175—212, 213—226. 
1870. 1. Lipscuirz. R. Entwicklung einiger Eigenschaften der quadratischen 
Formen von z Differentialen. Crelle 71, 274—287, 288—295. 
1875. 1. Beez. R. Zur Theorie des Kriimmungsmasses von Mannigfaltigkeiten 
höherer Ordnung. Zeitschr. Math. u. Phys. 20, 423—444, 21, 373—401 (76.1) 
and wes, w= 7, OS Oe) (79.1). 
1876. 1. Lipscuitz. R. Beitrag zur Theorie der Kriimmung. ,Crelle 81, 
230—242. 
1880. 1. Voss. A. Zur Theorie der Transformation quadratischer Differential- 
ausdrücke und der Krümmung höherer Mannigfaltigkeiten. Math. Ann. 16, 
129—178. 
1885. 1. Kituinc. W. Die nicht-euklidischen Raumformen in analytischer 
Behandlung. Teubner, Leipzig. 3 
_ 1898. 1. BEerzoraAri. L. Sulla curvatura delle varieta tracciate sopra una 
varietà qualunque. Torino Atti 33, 692—700, 759—778. 
1899. 1. Brancui. L.-Luxat. M. Vorlesungen über Differentialgeometrie. 
Teubner, Leipzig, 659 p. 
1902. 1. Ricci. G. Formole fondamentali nella teoria generale delle varieta 
e della loro curvatura. Rom. Ac. Linc: Rend. Ser. V, 11/4, 355—362. 
2. SERVANT. M. Sur une extension des formules de Gauss. Bull. Soc. Math. 
de Fr. 30, 92—100. | 
1) Beez, 79.1, p. 76; Vr in Br. 
