329 



nislied by the sum of all odd ones dixided by p! and may be 

 expressed as Cajlejan determinant: 



i ai . . . . a,, -u bjD^ 



1 ! . . <u Ci.S. ^ 



ai -- 32 ^ . . . . ^ a;. = ai . . . . 3/, = - - I • • i "? -2 'S | . (3) 



p!\ • • ^ ^ g ^ ^ ^ 



^-^ oj u o 

 ! 31 .... ay, -a rj *- 



The alternating product of /; fundamental elements is a linear 



quantity for p^n. For p^n it is zero. A symmetrical product is 



never a linear quantity. 



The Associative Systems R,i. 

 Classifying up to the lin. homog. group inclusive, the system 

 belonging to the linear quantities is //„, which is an associative systen), 

 entirely determined by the rules: 



e/ -< Qj — - ey h e/ = e;.; ') e'^ h e'j = - e'j h e'/ = e'/j 



e, -. Ci = k e'; h e'/ = k' 



e/ e_y . . . . e/ = e/j... / e\- e'j . . . . e'/ = e ',>•.... / 



ei2....>i = 1 e'i2....H = r 



ei = >£" e 2 e'„ I, ei h e'l = e'l h ei = ;:, e'l = x" 62 . . . . e« I' 



e,, . . . ., e„ are the covariant fundamental units, i.e. units of a 

 fundamental element, and e\, . . . ., e'„ are the contravai'iant funda- 

 mental units belonging to characteristic numbers, transforming 

 themselves contragrediently relative to the fundamental variables. 

 When classifying up to the equiv. group inch, the 'system Z^^ is 

 constituted, being obtained from the preceding one by the identification 



i = r 



and being entirely determined by the rules : 



e. ^ ef=k /, y, ....,/= 1, ..... /2 (4) 



eiC; .... tl = Qij ...A 

 ei2 . . . . n = I 



Quantities, whose units, apart from an eventual factor I, do not 

 contain two equal fundamental units as factors, exist uidike the 



') In a more exhaustive investigation "Die direkte Analysis zur neueren Relativi- 



tatstheorie", Verband der Kon. Akad. v. Wet. Sectie I Deel XII N'. 6 we consider 



C/C_y — Cj'Cy 



also not linear quantities and we write e/ ej = e/j and e/ -< ej = = Qij 



etc. For more convenience we write here e/n ej^^eij. 



