159 
eat Ae eas 
FeeFse—=(—1) ? P Pe HT Le HT LPT HS Lif tad 
iP nh ‚(81Ĳ) 
mutate (0, P) 0, P P) 40 Hebe Hh? Hote 24 Age 
in which TEN *g* may be replaced in each term by every arbitrary 
permutation of ideal factors of the u factors *g, each term now 
being on its own symmetrical in the last 2u factors. Since further 
2p 2u 5 
and A, is an alternation with w systems of two factors, we can 
build up every term of ove Gg by the ideal factors of u quantities 
H°:g and 2u quantities *g’. Hence ooge depends only on °*g’, 
g? 1K and K’ 
Hence we have got the following theorem. 
In a definite point P of a Vm in V, we consider the sums Cue of 
the a-factorical products of the principal curvatures with different 
suffices with respect to a definite normal direction i, of the Vm. 
We then form the mean values Cue and Oe a+tp24 of all 
quantities Oze TSP. OxzeGpe with respect to all possible normal direc- 
tions. Then One resp. Gre Ope vanish for a resp. a+ B odd, and for 
a, resp. a+ even they depend only on the linear element of the 
8 4 
Vn and on the Vy-component g+ K of the Rimmann-CurisTOFFEL 
4 
affinor K of the V,. Hence these quantities are invariants of defor- 
mation for a Vy in S, and for a Vy in V, they are invariant 
with me eventually possible deformations of the V,, in V, that leave 
8 
EK unaltered, in par ticular with all deformations with which the 
m-direction of the Vy», in P remains unaltered *). 
1) Lipscnitz has first proved (70, 1) the invariance of the cue for x even 
and for Vm in Rn, afterwards (76, 1) that of czecge for + even and 
20 +1. Kituinc has (85, 1) given the proof for all Cue and Cue The for), in 
Sn, he also gives the geometrical interpretation of these expressions. MAscHkKE 
has (06, 1) proved the invariance of oi for Voi in Ry with the aid of his 
symbolic calculus, Bates (11, 1) that of oc, «>1 for Voi in Rn. Bates has 
= 4 8 4 
not succeeded in expressing the c. in %’, K and g’4K for V in Vz, though 
he finds expressions for „… These, however, contain still the »—m functions that, 
equalled to zero, form th. equations of the Vm. 
