1922. No. 13. 



ON A SPECIAL POLYADIC OF ORDER n — p. 



(10) 



Ci e., 



e„ 



Ci Co ... . c„ 

 f'l 1 «1 2 • • • ^^1 " 



^Pl ^P2 • • • • ^^P " f 1 « f o » . . . . f H H 



Cl Co ... . e„ 



f J 3 f 2 2 . . . . tn ., 



Co 



r?;,,,. 



e« 



. ^r^; 



which shows that the space complement of any p vectors is invariant with 

 regard to any orthogonal transformation of coordinates (invariant under the 

 group of orthogonal transformations). 



Now let us assume that the p vectors a^ . . . . Cl/> all are expressible by 

 the same p unit vectors, i. e. : the /»-space containing cl^ . . . . Ckp also 

 contains p of the unit vectors, and we ma}' assume without loss of generality 

 that those are the first p vectors C^, C, . . . . e^». Then all the components 

 a,j vanish (or J ^ p and we evidently get: 



11) <,fa, . . . . C^p = (— D<" + 



I)/' 



e^ + i .... e„ 

 ^p + r . . . . e„ 



. . Û 



Ip 



Gp, 



app 



\. e. : the space complement is expressed by the other unit vectors (and a 

 scalar). This proposition is general. In other words: 



(a). The space complement of aiiv p independent vectors is expressible by 

 vectors lying in the {n — p)-spoce zvhicli is absolutely perpendicular to the p-space 

 contai)iing the p primary vectors. 



In order to show this it is sufficient to transform the p vectors into 

 a new rectangular coordinate system and to choose the first p unit vectors 

 of this system such that they are contained in the /»-space on the p given 

 vectors in question, which is always possible. This done the problem is 

 reduced to the case mentioned above (under (11)), and our proposition 

 is proved. 



It follows directly from the definition § 4 (a) that : 



(b). The space complement of any pennntation of a given set of vectors 

 is equal to the space conplement of the given set with the same or opposite 

 sign according as the permutation can be obtained from the given set by 

 means of an even or odd number of transpositions. 



