1922. No. 13. ON A SPECIAL POLYADIC OF ORDER ;/ — />. 3 1 



As we, by § 6 (a), have : 



(19) e^Xf.- ?<.Xe„ 

 we obviously get : 



(20) . e. X t' -= — e, X X, 



an equation which is well-known for the three-dimensional case*. 



A tew other properties of 0v, which are completely analogous to well- 

 known vector product properties in S.,, shall also be mentioned. 



The equation § 8 (14) is valid if we instead of bXc put a sum oi 

 such expressions. From this we deduce : 



(21) Ü X" - 2 0„ = — (« — 2) ! Ü • ((? — (Pc) 



analogous to the equation in Sg-. 



(22) 'cy<'F, = ~^-{'V— wx 



If we put Ü = e, in (21) we get the // equations: 



(23) e, X" -2 0-,= —{n — 2]\ (f, — y.,) 

 corresponding to the following three in Sg:** 



i X W, = - (a — x^) 



(24) i X ÎF„ = — (b - xJ 



fX ¥r = — (C — Xg) 



x^, Xo, x^ denoting here, of course, the conjugate system to a, b, C. 



§ lo. The reciprocal system and the „Ergänzungen" of 

 a given set of vectors. 



Let the reciprocal system, say f,*, to a given system f,- be defined (as 

 in .S.,) by the equation 



(1) r? f- = p' e<- = f' f'*- 



It is here convenient to introduce, as we have done in S^, the „Er- 

 gänzungssystem" of a primary system. f If the latter be a, b, C (in S.^), the 

 „Ergänzungen" are: \X\ = b X C, W.. = C X a = — a X C ; IV .j a><b; 

 Vl\ is the Ergänzung of a, W., that of b, etc. The reciprocal system of 

 a, b, c is, as mentioned § 3 (6), obtained from the „Ergänzungssystem" by 

 division bv the determinant of a, b, C. 



' Almar Næss: loc. cit. § 4 (4I. 

 •* loc. cit. § 4 (2). 

 t loc. cit. § 13. 



