Mathematics. — “On expansions in series of covariant and contra- 
variant quantities of higher degree under the linear homogeneous 
group.” By J. A. ScHouren. (Communicated by Prof. J. 
CARDINAAL). 
(Communicated in the meeting of March 29, 1919). 
Notations’). A covariant affinor of degree p may be written as 
the general product of p ideal fundamental elements’): 
Pp Marts fe 
Wer ee aa, BOL ug ei U(L) 
yaad oe i p 
P 
an alternating or symmetrical one as a power of one ideal funda- 
mental element : 
1,...,0 
pv = = PU), Oi == WP, pen 
hed 
2 
lr ( ) 
BW ae PEN OLENE 
Aare Al 
. P . : . . 
The p! isomers of u, viz. the p/ products of the ideal factors in 
pP 
all possible orders are real rational covariants of u. Each isomer is 
P 
formed from u by one of the p/ permutations P, of the ideal factors. 
N dits : Dat 
By a penetrating general product o of some affinors u,v,... we 
pq ; 
understand any isomer of the general product uv.... An affinor, the 
different isomers of which are not connected by linear relations, is 
called a non special one. 
Classes of isomers’). It is well-known, that the p factors of an 
isomer can be divided into groups of s,,s,,... factors in one single 
way, so that in each group the permutation is a cyclic one. The 
groups are called the permutation regions and the complex of the 
numbers s,,s,,... in descending order and omitting all numbers 
1) See further “Die direkte Analysis der neueren Relativitätstheorie.” Verh. 
der Kon. Akad. v. Wet. DI. XII NO. 6, p. 7—11. 
*) Introduced firstly by E. WazLscu under the name of “symbolische Vektoren” 
in “Ueber mehrfache Vektoren und ihre Produkte, sowie deren Anwendung in der 
Elastizitätstheorie.” Mon. f. Math. und Ph. 17 (06) 241—280. 
Er 
Proceedings Royal Acad. Amsterdam. Vol. XXII, 
