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



37 



We can without loss of generality assume that a stands for the first 

 and the second rows, or respectively columns. Then more explicitely we write 



the equation which we have to prove, thus : 



(3) 





The symmetric differences can be expressed as the scalar values of 

 all the two-rowed determinants — taken with the sign — ( — 1)' + ',/ and / 

 being the two columns represented in the determinant — of the fol- 

 lowing matrix : 



ÙÛ f" 



Cj e.. e« 



(B) 



i. e. : we have to take the scalar product of each two vectors to be 

 multiplied. But we also notice that the symmetric differences in the same 

 way can be formed from the matrix . 



(C) 



Cj e, 





X, being the conjugate system of the f s. 



Now all the quantities '5//>i-2 ^^'^ ^^^ the in — 2)-rowed determinants 

 of the matrix : 



/l 8 . A i /i " 



f"J'n, fnu 



obtained from the matrix of the f s by striking out the columns 1 and 2. 

 And in order to form "5/ / - 1 2 ''';■ ' ^^'^ have to multiply each d\i by the 

 corresponding one of these determinants and add up all the products. 

 But then we see that this sum is simply got as a determinant, obtained 

 from iDi by replacing the two missing columns by the matrix (Hi, whose 

 two-rowed determinants — as said above — exactly give the quantities 

 r/// as their scalar values. Changing rows and columns in this determinant 

 we thus obviously have : 



f„- 



. . . . e„ 

 .... J, I <{ 



(4) 



(J') 



ti • r., • 



./13/2 .'5 



A"Â 



fn 



The validity of this equation is also readily shown by expanding its 

 second member in terms of the two-rowed determinants of the first two rows. 



