828 



Application of Klein's l^rinciple of Ciassijication. 



Tlie author of this [)a|)er observed in 1914 ') that it follows from 

 the a|)|)li('atioii of Klein's principle of classificatioii to analyses 

 belonging to definite quantities, that to a given group of transfor- 

 mations and given quantities belongs a coinpletelj deteiniined 

 system, which may simply be conq)uted. This was practically done 

 for n = 'S, the rotational group, and quantities up to the second 

 order inclusive. In a more exhaustive investigation contemplating four 

 different sub-groups of the linear homogeneous group the same was 

 executed for arbitrary values of u and for (piantilies of an arbitrary 

 degree"). We shall briefly state some results of this investigation 

 bearing on linear (piantities, in particular for // ^ 3 and 7?. = 4, 

 founded on the : 



rotational (/roup (a,' -j- • • • + ""' invariant, det. = -f- J) 

 and availing ourselves of the: 



orthogonal group {a,^ -\- . . . -\- a,^ invariant, det. = ± 1) 



special-a/jin. group (lin. hom. with det. -\- 1) 



equivoluminar group (lin. horn, with det. ± 1) 



linear homogeneous group 

 forfurtherclassificationofthequantitiesexisting with the rotational group. 



General si/inmetrical and alternating multiplication. 



Three multiplications of fundaujental elements exist with all the 

 sub-groups of the linear homogeneous group au'l for all the values 

 of n, viz. the general, the symmetrical and the alternating one. 



The general product of p fundamental elements has nt' characte- 

 ristic lunnbers, being the products of the characteristic numbers of 

 the factors. Their mode of transformation is entirely determined by 

 this defiuiliou. We express the product in this manner: 



o 



ai o a-j o . . . , o ay, = ai . . • . a^/ (1) 



o 



By isomers of a, .... a^, we mean all the general products that 



o 



can be formed by permutation of the factors from a^ . . . . a^,. An 

 even respectively odd isomer is concomitant with an even resp. odd 

 permutation. The sgmmetrical product of a, ... . a^„ is the sum total 

 of all the isomers divided by their number p ! : 



w 1 ^ o 



ai ^^ a'i ^' . . . . -^ ü/j = ai . • . . ^y, = — -2" a,, . . . . ai/j . . (2) 



The alternating product is the sum of all the even isomers dimi- 



^) Grundlagen der Vektor- und Affinoranalysis, Leipzig (14). 

 2) Ueber die Zahlensysteme der rotationalen Gruppe. Nieuw Archief voor Wiskunde 

 1919. 



