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



39 



In order to get an expression for the sum ^ <5i2j rtdrt, we can proceed 



(rO 



in a completely analogous way. We get : 



(10) 



Krt) 



e„- 



er e.,- 



y-x '^=1 >^« 



J-A\/-ii ./3 " 



J II 1 J II i Jim 



readily seen by expanding according to the two-rowed determinants of the 

 first two rows li. e. according to the quantities r/,-/), because we now shall 

 combine d, t with determinants of that matrix which is obtained by striking 

 out the first two rows of the matrix of the f s. 

 We here put: 



(11) xr = ^/n^n y-, = ^.Aoe, 



and inserting this in ( 1 ) we get : 



x>i — ■—_//'« Ci 



12) 



(r/) 



e^- C, • Ç,r 



./31 Å-2 



Jill Jii-i- 



/It H 



and this determinant can be reduced to the sum of the // determinants of 

 the following form (/ = 1,2 n): 



(131 



Ci" e., • e„- 



/,i e,/,., e, /,« e, 



/31 /s 2 ^3 " 



/11 1 fn 2 J I' I' 



Ci e._. e„ 



/1/0 /ii: 



/31/3-2 Js" 



./«1/- 



/>,« 



But if we here put / >> 3, we get vanishing determinants. Therefore 

 we have : 



(141 2'(5i.,. rtdrt = 



/xi/12 

 y 3 1 /3 2 



/«l/"2 



e» 

 ./s" 



J II It 



ei- 



./21/22 

 /3 1 y 8 2 



/2" 

 ■/3" 



/" 1 ./" 2 



/.... 



eo 



