278 
term. That such an expansion is possible and singly determined has 
been first proved by W. Gopr’). 
Thirdly it is possible to decompose each elementary affinor into ordered 
elementary affinors of the second kind. To this corresponds the singly 
determined expansion : 
Le Cs eer 
Fu Sey BMA mx ys 
see. we E P 
EK. Wartscn?) has given another expansion which is also singly 
P 
determined and which corresponds to an expansion of m in terms 
of the form 
with coefficients that for a definitely chosen order of u?, v’,.... 
are functions of a,,a,,... It is remarkable that the number of 
terms of this expansion for a P-linear form is equal to that of the 
expansion with regard to ordered elementary aftinors of the first 
kind, eg. 1+5+9+5= 20 for P=6. 
7 . 
Kepansion in series of a form in m sets of n variables. 
Let 
P 
FA MEN en NPN Ee 
P 
-P 
be a form in the m sets of variables x’, y’,... and m= wry’... 
P 
the corresponding affinor. We can expand m in non-reducible 
covariants. Each term is then a sum of ordered alternations 
each consisting of a penetrating general product of a number « of 
factors E that is characteristic of that term with a linear homo- 
geneous non-reducible affinor of degree P—an. To this corresponds 
') W. Gopr deduces this expansion in quite another way and this may 
be the reason that he has not seen the connexion with the group characters of 
Frozenius and the possibility of an analogous expansion for n > 2. “Ueber die 
Entwicklung binärer Formen mit mehreren Variabelen”, Arch. f. Math. u. Phys. 
fe DSj Ald: 
*) Ueber Reihenentwicklungen mehrfachbinärer Formen. Sitz. Ber. der Wiener 
Akad. 113 (04) 1209—1217, Warrscm has used for the first time the expansions 
in series of the theory of binary invariants to decompose directed quantities in 
parts covariant under the orthogonal group (e.g. the decomposition of the affinor 
of deformation in scalar, vector and deviator), ‘Ueber höhere Vectorgröszen der 
Kristallphysik etc.” Wien. Ber. 113 (04) 1107—1119; Extension de l'algèbre 
vectorielle etc., Gomptes Rendus 143 (06) 204—207. 
