154 
respect to the V, is the sum of the relative curvatures with respect 
to these Vn *). 
§ 5. The relations of the relative curvature to the principal 
radu of curvature and the simplest invariants 
of deformation. 
For m= n—1 we have: 
9 
ns rivals 4 ~ a 
K, = — TE Tk (h — 'h) (h ~ ‘h) 
2 
SS SS SS =a ig id it da 4 (h di 'h) (es is 'h) 
KC dou) 34 4 h'hh'h 
ot TE ia ip (io — ia) 4 
i 
= eS (ints? Vidi Gila) Gaia? VA) Mi Vi) 
m (m—1) a 
Choosing the i, in the principal directions of curvature, we have: 
ide Wii Dn et 
. . 2 Vv . 1 (49) 
la la 5 n— ee en 
i E,, 
and we get: 
afb 
1 Er al (50) 
~ m(m—1) a5 Ra Ry 
in words: 
The relative curvature of a Va ina V‚ is the mean value of 
the twofactorical products with different suffixes of the principal 
curvatures. 
Hence for a V, 4 in S, this sum is an invariant of deformation, 
for a V,1 in V, with K,—=O it is equal to the negative forced 
curvature. 
For arbitrary values of m we have: 
1 sha ay 
K, =— ———_ > Cais ® V ic) (isia 2 V ic)N—(ia in? V ic)(is is 2 V ie). (61) 
ESET € 
Now choose the i, in different ways for each value of i, and each 
time in the directions of principal curvature with respect to the 
normal vector i,. 
1) Kruuine, 85, 1, p. 247, Vn in Sn; BerzoLar: 98, 1, p. 697, Vm in Sn, (both 
authors use projections); Ricci, 02, 1, p. 361, Vm in Vn. 
