156 
When we want to write out the powers, we have to introduce equi- 
3a 
valent factor systems HHS, ‘H,, ‘H,,’H, etc. H is symmetrical 
in the last « factors, just because all « systems are equivalent, and 
contains in these factors only the units i,41,...,1,. We then have: 
3a 
het he = He if (56) 
and: 
ae) on 3a 
Oue =(—l) 2 ge he ig. (57) 
The mean value 6,, of the values of 0, belonging to all possible 
2a 38a 
directions i, can be obtained by transvecting g’?%H with the mean 
value of all quantities is. Now it can be easily calculated that this 
latter mean value vanishes for « odd and >1, and is equal to 
(?g—’e"\"~ save a numerical factor for u even, a= 2u. So we have e.g. : 
ani 
M.value ie ie ie ie = ge gS 58 
value ie icici RATE kn _ (58) 
15 vel 
M value i. = (*g—'*¢')3-. (59) 
(n—m)* + 6 (m —n)’ + 8(m—n) 
3a 
Since H-is symmetrical in-the last « factors; we thus have for « = Dye: 
4u 6h : = 
One = 2, $4 HH gr | | (60) 
where u is a positive integer, 4, a numerical factor only depending 
on n—m, vand in which expression *g* may be replaced by every 
permutation of the 2u ideal factors of the u factors *g. 
6 6 
Now H ge can be built up by the ideal factors of H ?*g, but 
since on account of (37): 
6 Berk ad 
OH2*g——g4K4+K, — (61) 
6u 8 4 4 
we see that also H 24 *g# ony ganes on g’ 4 K and K’. Hence 
6x Only depends on 7g’, est 1K and K’. Consequently it has the 
same invariance as K,. 
Now choose «+ 8 even, « fi 8 = 2u and consider the quantity 
6 
VS ean eee (62) 
32 
built up of H and H, L being equivalent with H. When we want to 
write out the powers we have to introduce « equivalent systems 
H,, H,, H,, ‘H,,’H,,’H, etc. and in the same way # systems 
BLS, Ly, LL, ete. We then hares 
