1922. No. 13. ON A SPECIAL POLYADIC OF ORDER 11 — p. 



23 



But, by § 7 (5), all the space complements lO vanish where tj is 

 equal to one of the unit vectors in the determinant. Hence it is sufficient 

 to take into account those terms only where e^ is equal to one of the 

 vectors Cyti, C,t2 • • • ■ 'ik , which are stricken out when forming the deter- 

 minant. For each set of the /"'s we thus get only p terms of the form iC). 



Let us consider a fixed set of the k's and form all the space comple- 

 ments with regard to this set. The first one will be : 



e-t, X" 



ei 



(7) 



,e k: . . . e„ 



[n-pV 



e„ 



[ii—p)\ Cam X"- ''I ^1^-2 • • -//t,- . .e„) 



e, . . . . Çk\ . 



ex-, 



e„ 



Each of the last ;/ — p -\- 1 rows, being components of a unit vector, 

 consists of 1 and // — 1 zeros. Of all the determinants which can be formed 

 from these rows there is therefore only one which is diff"erent from zero. 

 The sum of the indices of the columns of this nonvanishing determinant is 



2 Î 



and the sum of indices of the rows is 



(p -^ n] \ii — p + 1) _ ir ^ p~ ^ II + p 



Expanding (7) after Laplace according to these determinants of the 

 last n — p + 1 rows, we thus get only one term, the following : 



■ " — p (J/ — I) H 



P 



(-1) 



(D) 



[II— py. 



ex-2 . 



tk2 . 



^k. 



ex-. 



We must especially notice that the columns stricken out of the last 

 n — p + 1 rows to form the second determinant of (Di (the last factor of the term) 



