WILSON' AM) LEWIS. — RELATIVITY. 437 



changes of sign. In the final form thus found tlie subscripts g . . . 

 and j . . . ha\i' successively to he canceled. Hut one of these is 

 necessarily the subscript 4 (corresponding to the (5)-vector), which 

 re(|uires a change of sign. Hence 



{kg...'ki...)'ki... = - (- iy^''-"%..., 



and the desired result is proved. 



Consider the product a**,?*. We have by (24) cither 



a*..9* = (a*x,9)* or ,3*-a* = 05*xa)*. (27) 



Now, although a**/?* and ,9* "a* are equal, the two expansions obtained 

 are usually diti'erent. In fact, as the total dimensionality of an outer 

 product cannot exceed u, the first formula holds only when p ^ q, 

 and the second only when q ^ p. Let us assume q ~ p. Then 



a*. ^3* = .3* -a* = (,3*xa)* = (— l)P(«-9' (ax,3*)* 



= (_ l)p(n-g)a.l3** = — (— l)^s-P)^''-«'a.;3. (28) 



As a corollary 



a^'tt"^ = — a* a. (29) 



The complement of an inner product may likewise be proved to be 



(a./3)* = (_])P(n-p)ax/3*, (30) 



where it is assumed that the product a*.? has been so arranged that 

 the second factor is of dimensionality q greater than the dimension- 

 ality 2? of the first. We haAc furthermore 



a*xa=(a.a)*; (31) 



and also if /? is a pseudo-scalar 



(a. ,9)* = (~ l)P^-~P\3*a = [3 -a*. (32) 



It is important to observe that by means of these rules it is possible 

 to replace any outer product by an inner product, and vice versa. 



33. We are now al)le to obtain rules for the expansion of the vari- 

 ous products in which three vectors occur. The simplest type, and 

 one which needs no further comment, is 



(ax^)x7 = ax(^x7), (33) 



which follows from the associative law. 



