269 
We have therefore, when for the sake of simplicity, is written . for. : 
1 
Mm. =m. =mAr =x (ma es Se et ee ae ) 
1 1 1 n n 
: p q 
By 7-th outer transvection 9 of m affinorsu=u,...u,, V=V,...V 
; vq 
etc. will be understood the quantity. found from uv by substituting 
locally for the ideal vectors u,,v,,..., and then for u,,v,,..., 
to U;,V;,... the ideal factors of their alternating product. When 
at the same time the other factors are locally substituted by the 
ideal factors of their alternating resp. symmetrical product, then the 
i-th outer alternating transvection A resp. the i-th outer symmetrical 
fi 
one V is formed. 
2 
p 
Affinors and algebraic forms. When the P-th transvection of m 
P 
is formed with a product r’ of P different contravariant fundamental 
elements r,,....,%p, we find the form: 
Pwop 
ERE PE aici (Me BP) 
vel Of 
EP eee En, 
RE bne Hemi riet koe bd Ep 
1. Pp 
. . . B . 
A special case is that Tr’, ....r’p are all not-ideal. Then Fm is 
a form in P sets of n not-ideal variables. Thus the characteristic 
numbers of a covariant affinor (and therefore also of a contravariant 
one) may always be considered as the coefficients of such an alge- 
braic form. When the sets r’,,...,%p are given and when their 
P P 
order is fixed, Fm is singly determined by m. When all sets are 
P P 
different, then m is also singly determined by Fm, in the other 
P P Pp 
case not, as m-++n, where n is an arbitrary affinor, alternating in 
two factors that correspond with two equal sets of variables, trans- 
P P 
vected with r’, also forms Pm. 
In the general case r’,,...,¥rp are ideal, 4 is however equal to 
BA Dente 
X', Ys .…., where po+go+...=P, and where x°, y’,... are not-ideal. 
Then Fm is a form of the degrees 9,6,.... in sets of variables 
that may be considered themselves as coefficients of variable forms 
in p,qg,... setsof n covariant variables. Such variables will be called 
pq 
variables of the degrees p,q,... When the sets x’, y’,... and the order 
18* 
