1922. No. 13. 0\ A SPECIAL POLVADIC OF ORDER // — p. 29 



Let a dyadic in S« be defined as ^ = e, f, (sum for / as usual fi-om 

 I to ;/). Then the quantity wliich is analogous to Wj of vSg, must be: 



17) 



0. = e, X f, 



i. e. : a polyadic of order // — 2, the space complcuicnt of 0. We obtain a 

 formula for 0^. b}- putting f , = e, in (6). Thus we get : 



(8) 



0. = e,Xf, = 



Ci e, 



c„ 



ei e,. 

 Ci ■ e, • 



y.» 



But as : 



(91 



^j-'''-i =fji = «/-fy 



we can write : 



(10) 



0-. = e, X f, = 



e, Co 



e„ 



ei e, 

 e, • e, ■ 



e„ 



Here we have for an}- J and / \J <C /j : 



(11) 



ey C/- 



f. f/ 



= e,-f/-e/-f; = -(/y7-/o) 



e„ 



If we develop (10) in terms of determinants of this kind, the sign of 

 (111 will be (— r)«-i+"+y-/= — (— 1)./-^. Let us by Ej i denote the 

 [I/ — 2)-rowed determinant defined by the unit vectors after erasing e/ and 

 e/(y< /I, i. e. : 



Ci . . . /j . . . /i 



(12) Eji 



i. e. : The E's are defined by the equation 



(13) H-i-l)'-^' Ej 

 Moreover, we put : 



C/ e/ 

 ey C/ 



e» 



C/, 



14) 



(- l)'''(/,7 -//>) = rt'y/. 



