1922. Xo. 13. 0\ A SPFXIAL POLYADl'C OF ORDER // — />. 43 



That is, the matrix of A' is determined by : 



By the definition of the y"' Erganzungsvector, § 10 (a», this means: 

 I/I AVy is obtained from the determinant 



./ 1 1 ./'2 1 



f'n 

 f". 



J\" J-i " ./"" 



by interchanging the y'^*' row with the quantities f\,, f »i, .... /"'„,. Or, 

 in other words : 



l/j AV; is obtained from the determinant of the f s by interchanging 



its y'' coUimn with the /''' cohimn oi the determinant of the f 's. 



§ 14. Miscellaneous Formulae. 



The space complement of a set o'{ p unit vectors must, by § 5 (a), be 

 expressible by the other up unit vectors. Let the set be Cx-i, Cx-a • • • • Cx- : 

 We will assume that they are arranged in order of magnitude, i. e. . 

 k, < k., < . . . . < k.. 



(1) 



<f ex-i e/,-,. 



ex- 



Ci . . . . ex-, 



P(r-n , " 

 " P ^ r 1/-, 



ex'.. 



ex-, 



C',-. 



. e„ 



e« 



e« 



C;, 





 



where, as before, Cj . . . . /k- . . . . e« denotes the set of the n — p unit 

 vectors which are left after erasing the Cx/s. This formula taken into 

 account, we can write § 8 (6) and (A) : 



(2) 



<'^ a^ a.> . . . . up = 2l (p e ;, c,;. 



^1 X-, 



(1r /. 1 



^?i K\ 



<jp k\ 



