271 
As the transvection 
ei, Ci DOT Cp IE 
DE =de An 
ie -equal fom for the case that == jets P and equal to 
zero in every other case, we have 
ep 
ip 
q 
F=n.xy... 
P 
P Pq ; 
When now on one hand n and on the other hand x’, y’ are written 
as the products of ideal fundamental elements, then /’ is really 
painced to a produ! of P ideal linear forms. In order to derive from 
n the affinor m that is singly determined by /' we first prove the 
theorem : 
Theorem 1. 
P 
When q is an ordered elementary affinor of the first (second) kind and 
P 
r’ a ditto one of the second (first) kind’), then the P-th transvection of them 
is zero, when the two elementary operators gij A?) HD, Ein MM (8) AD, by 
P PP P 
means of which q and Y’ (£° and q) can arise, are not conjugated, 
ie. when not /=j, m=ianda=@. 
As: 
re Ei j AS MS ie dn Elm Mm af Dt 
we have 
P P ( = P P 
(4) (a) » 
q pr = ij tm AP MG a), Min Age 
(3) 
4” (a) ) 
— Ei j Elm (u; 4 A; M; An Af — 
\ P P 
(B) rl) le) (a) ) 
= Bij Elm (at Mi A; M; or : 
Thus the transvection is in fact zero, when not l=j (therefore 
also m7) and «=p. The same proof holds m.m. by changing 
the first into the second kind and vice-versa. 
Let now the sums of the ordered elementary operators of the 
° p q 
first kind that do not annihilate x’, y,...., be ,L,,bL,.... and 
the sums of the conjugate operators *L, /L,.... ete. In the special 
case that „ZL is a sum of elementary operators we evidently have 
P P 
~L—=h. From n we first form an affinor n,, by permutating the 
1) See Ey p. 262. 
