155 
We then have 
1 afb 1 1 
"~~ m(m—1) eo anu Bia Reh” 
(52) 
in words: 
The relative curvature of a V in the V, is the sum of the mean 
values of the twofactorical products of the principal curvatures 
with different suffices with respect to n—m arbitrary mutual perpen- 
dicular normal directions of the Vn. 
Thus for a V,, in S, this sum is an invariant of deformation and 
for a V,, in V, it is invariant with all deformations with which 
Bait 
g’*K is invariant in P, in particular with all deformations with 
which the m-direction of the V,, in P remains unaltered. For a V,, 
in the V, with AK,=0O it is equal to the negative forced curvature. 
§ 6. Other invariants of deformation of a Vm. 
Now we consider the sum of the four-factorical products with 
different suffixes of the principal curvatures with respect to the 
normal i,. Since 
8 8 
sg § he if” = ¢ 8 (he—'he— "he a i.) (he — ‘he — "he a he) == 
== inti leigGgicdsin)* he hohe hehehe heh = 
abcd 
abed fF (53) 
Pat a (ia ia 2 V ie) (ie is 2 V ie) (icic 2 Vie) (iaia? V ie) — 
abcd x 1 
ENT te ve, 
abed Rea Res Rec Red 
8 
this sum is equal to g'Sh, h, . In the same way we can prove 
that for the sum o,, of the «-factorical products with different suffixes 
of the principal curvatures, a <m, with respect to the normal direc- 
tion ie the equation holds: 
a(a—1) oa 
| Gan =(1) % EERE hee, (54) 
We now form the series of quantities: 
6 
H = (H, ~'H,) (H, —'H) H, 'H, 
9 
H = (H, EN H, vrt "H,) (H, ‘Tt; 'H, oF, "H,) H, H, "H, (55) 
32 
= nt BT 
