1922. No. 13. ON A SPECIAL POLYADIC OF ORDER H — p. 



13 



For brevity we will denote the space complement of a^, a.> .... a/) by 



(c) </> a^ a.. ... a/, or : a^ a.. . . . a/, />> 



Hence the operation sign </> or /■> indicates that /> vectors written 

 to the right, or respectively to the left, shall be combined into their space 

 complement. If we are going to derive the complement of s + / vectors, 

 s to the left and / to the right, we write -X', e. g. : 



(1) 



ai a.2 <\ 2X3 1\ h. bg b^ b-^ = a^ 



evidently a polyadic of order 1 + (// — 5) + 2 = ;/ — 2. If s and t both 

 are equal to one, we write X- Thus the space complement of two vectors 

 a and b may be written : 



(2) aX 6 = <2ab = ab 2> 



which in S^ coincides with the ordinary vector product of a and b. 



§ 5. Invariance with regard to orthogonal transformations 



of coordinates. 



First we will show that the space complement of any number (say p) 

 of vectors is independent of the particular (orthogonal) coordinate system 

 which we may choose : 



Let 



e'l, e'.,, . . . e';, 



be a system of orthogonal unit vectors, defining a new coordinate system, 

 defined by : 



(1) f'l = ^i^xi', e'.2 = C,f2'; • • • • e'« =-^e,£„,- 



where consequently 



(2) Ej^^ + Ej.} + . . . .+ Ej,? - 1 for all y s 



(31 and e,\ Ej\ + f/.^ fy.j +....+ f,« ej,, = i ^ j 



Further, let the components of the vectors iX^ , a., .... a„ with respect 

 to this new coordinate system be primed, such that for any j 



(4) 



a,- = e , (7 



