Mathematics. — "On the direct an<(hjses of the /iitear qumititie.s 

 helong'uui to the rotational (irou[) in three and four fundamental 

 variahh'.s\ Bj Prof. J. A. S(jhouten. (Conininiiicated bj Prof. 

 Cardinaai,). 



(Communicated in the meeting of September 29, 1917». 

 Quantities and direct analyses. 



By a (geometric or algebraic) quantity existing with a definite 

 transformation-gronp we niean, according to F. Klein, any complex 

 of numbers fcharacterislic numbers of the quantity), that is transformed 

 into itself^) by the transformations of that group. Quantities only have 

 any signification and only exist with definite transformation-groups 

 and may be "disturbed" as such with othei- groups, whose trans- 

 formations do not transform the characteristic numbers into themselves. 

 They are completely (letei-mined by their mode of orientation, -i.e. 

 the mode of transformation of their characteristic numbers. The 

 vai-iables of the group are called fund<nnental rariah/es and are the 

 characteristic numbers of a fujtdamental element. If the group is 

 the linear homogeneous one in n variables, the sijnplest quantities 

 are those, whose characteristic numbers are transformed as the 

 deteiininants in a matrix of p fundamental elements independent of 

 each other, p = 1, . . . , n. With a homogeneous interpretation of the 

 fundamental variables they correspond to the linear /?„ -^rfomplexes 

 in /i,,_i, provided with a number-factor. All the quantities, whose 

 characteristic numbers are transfonned in that way under the trans- 

 formations of the rotational group, we call linear cfuantities. 



By a direct analysis we mean a system of an addition and some 

 multiplications by means of which we can express the relations 

 among quantities of a definite kind left invariant under the trans- 

 formations of a definite group. Every tpiantity is in the analysis a 

 higher complex number. Till recently suchlike analyses were brought 

 about by choosing for multiplications some characteristically distri- 

 butive combinations conspicuous in geometry or mechanics, and 

 ujiiting them into a system as well as might be. Owing to the great 

 number of existing combinations of this kind arbitrariness could not 

 fail to arise, and this led to the formulation of many systems, the 

 adherents of which have been involved in a violent polemic for 

 these twenty five years. 



1) e. g F. Klein, Elementannathemalik vom liüheren Standpunkte aus. Leipzig 

 (09) II p. 59. 



