20 



AI. MAK NÆSS. 



M.-.\. Kl. 



Now is 



(1) 



<r cli Cl.. . ... Cif 



And, by § 5 (b), the space complement of each of the even permuta- 

 tions of cli Cln . . . . up is ecjual to ^r Ci^ Ci., .... up, but of any odd permuta- 

 tion equal to the same quantity taken negativeiy. But the odd permutations 

 have, in the developed determinant, minus sign, which reverses the sign. 

 1. e. : the space complements of each term of the determinant in question are 

 — the sign of the term taken into account — all equal to the space comple- 

 ment of the principal diagonal. 



Thus we get: 



a, . . . . a, 



= /) ! <r a, . . . . ap 



(2) 



<^ 



^\ 



We get a similar result if we expand the space complement of a 

 polyadic of the form (order being /> + 1) : 



(3) 



(4) 



a, 



That is: 



<P + it) 



a, . 



<\ 



which we also can write :■ 



(5) vX^ • • 



Ci, . 



P< 



/ ! <f - 1 ï» ill 



pivXp<\ 



ein. 



We readily see that this quantity vanishes if ï» is equal to one of the 

 a's or, in general, linearly dependent on the a's. 



§ 8. A generalization of the expansion for the vector triple product. 



An equation which in ordinary vector analysis is of importance on 

 account of its frequent occurrence, is that of the vector triple product. In 

 quaternion notation it is written : 



