1922. No. 13. ox A SPECIAL POLYADIC OF ORDER « — p. 21 



(1) /\i(/'b CI == c5ab — 6 5c a * 



which equation Gibbs writes 



(2) aXibXc) = - a-(bc-cb]. 



It may be found more convenient, in this and similar equations, to 

 write such d3'adics (and also triadics etc.) in determinant form, as thereby 

 srreater svmmetrv is obtained : 



(3) 



aX<bXc) = — a 



In this form the equation can be generalized to //-space. It must only 

 be kept in mind that 6 X ^ in S,, is not a vector, but a polyadic of order 

 ;/ — 2. The vector a and this polyadic then combine to form the final space 

 complement oi (3 1. We can then prove that in any space Sn the following 

 equation is valid: 



(4) 



aX"-MbXci 



2)! a • 



But this equation can be still more generalized. We are going to 

 show that it holds good, not only for the triple product, i. e. : when we 

 have to derive the space complement of two vectors 6 and C and then 

 combine this with a, but also in the case when we instead of 6 and C have 

 any set of p iudcpcndoit vectors: a^, a., . . . . a^, (/> <C '/)• Üf the vectors 

 are dependent the theorem is true, but trivial.) Hence, the equation which 

 we will consider to be the generalization of the expansion for the vector 

 triple product, and which we now are going to prove, is : 



(5) f X "-/^ '<p ^■*i ^i 



-1-1) '{n-p)\^ 



. <x. 



11 being the number of dimensions of the space considered. We can tell 

 at a glance that it gives (4) as well as (3) as special cases. 



In order to prove (5) we first expand <p a^ a., . . . . Äp. By definition 

 we have : 



• As will be known, Ta 5 = — x b, Sû b = — a • b. 

 *' See: Zur Theorie der Triaden von Almar Næss (24), p. 108. 



