Mi) 



said products are doubled l)y introducing the "dual" Itivector (dualer 

 sechervektor) '). E. Wilson and G. Lkwis lune furthei' elaborated 

 the system and obtain all the products, but three"). All these 

 conclusions are founded on analogies vvilh the common vector-analysis 

 and the multiplications form no parts of the associative midiij)li('ation. 

 Therefore the free calculation-rides cannot immediately be put down 

 from memoi-y according to the trans\ection-rule, but in so fai' as 

 they exist they only allow a use by means of a table. The names 

 scalar and vectorial loo, have been di\ided over the existing 

 multiplications by analogy and not in agreement to the duality « — y. 



Wilson-Lewis 

 -haX b 

 a . b 

 + aX2b 

 -I- a . 2b 

 ±^a = ±aA: 



al 



aX b = 2C 

 a . h= c 

 a X 2b — 3C 

 a . 2b = c 

 = — I a =3b*) 



SOMMERFELD, LaUE, etC. 



fa b], vectorial product 



[a b], scalar ,, 



Ic= [a 2b*], vect. pr.w. dualbivect. 



— [a 2b], vect. pr. 



-h aXab 



- a . b 



-(- 2a X 2b 



— 2a . 2b 



a . 3b = 4C*) 

 a X 3b = 2C 

 2aX .b-.ic*)j 

 2a * 2b = 2C 

 2a . 2b = c 



± /: 2a = ± 2ak >a I = 1 2a=2b*) 

 kk= - \ 3 12 = + 1 *) 



I ^c = (2a 2b'^), seal. pr.w. dual biv. 

 [2a 2b], vector pr. (G. Mie) 



— (2a 2b), seal. pr. 



— 2b = + 2a* 



± /r 3a = ± 3a A: 



- aa . 2b 

 -l-3a. b 



3al = - ba^b*) 

 3a . 2b = 3C 

 3a X 2b = c 

 3a . b= c 

 3a X b = 2C 



b This is not a proper duality, because in the only duality existing with the 

 orthogonal group, a-y, a bivector e.g. e^j is not dualistic to the "dual" 

 bivector lojg, but to e^^ itself. 



') The connection with an associative Clifford algebra and the absence of 

 three products has already been briefly pointed out by J. B. Shaw, "'The Wilson 

 and Lewis Algebra for Four-Dimensional Space" Bull, of the int. ass. for quat. 

 (13) 24-27. 



