436 PROCEEDINGS OF THE AMERICAN ACADEMY. 



We may next show that 



(ax,5)* = a-/3*. (24) 



Suppose again that a, /? are unit vectors k^..., k/,... . We have to show 



(kg.,.xkA...)*ki... = kg...«(kA...-k]...), 



where ki... denotes the unit pseudo-scalar. Without changing this 

 equation, it is possible on both sides to arrange at the end, the sub- 

 scripts of the pseudo-scalar ki... in the same order as in the factors 

 kg..., k/,... Thus we have to show that 



{kg...-x'kh...)'kj...g...h... = kg...'(kh...'kj...g...h...). 



But now the products on the right are found by canceling succes- 

 sively the common subscripts h . . . and g . . .; whereas the product 

 on the left is found by canceling simultaneously the subscripts of 

 kg...k... . The identity is therefore proved. 



As a corollary of the two preceding results we may write the formula 



(ax/?)* ^ (— l)P9(/?xa)* - a./?* = (- l)P3/?.a*. (25) 



All these rules are true for any space, Euclidean or non-Euclidean. 

 The complement of the complement of a vector a is the vector 

 itself, except for sign. If a is of dimensionality p in a space of n 

 dimensions, the exact relation is 



(a*)* = — (— l)P^''-P^a. (26) 



The complement of the complement of a vector will therefore be the 

 negative of the vector except when p (n — p) is odd, that is, when the 

 dimensionalities of the vector and of the space are respectively odd 

 and even.^'^ For the proof, the consideration may be restricted to 

 the case where a is a unit vector k^... . Then 



(a*)*= (k,....k.-...).k,-... = (k,....k,...,...).k,...,... 



= (_l)P(n-P)(k,....k,...,...).k,...;... 



Here again the subscripts in the pseudo-scalar kj... have been re- 

 arranged so as to bring g . . . to the end. Then as g . . . denotes p 

 subscripts and j . . . denotes 7i — p, the permutation involves p{n — p) 



27 In Euclidean space (a*)* = (— l)^''*~^^a. Some writers who have identi- 

 fied vectors with their complements have perhaps overlooked this relation 

 which would, upon their assumption, make a vector sometimes identical with 

 its own negative. 



