170 PROCEEDINGS OF THE AMERICAN ACADEMY. 



From these rules ^° we obtain the equations, 



axa = 0. (15) 



axb = — bxa = {aih.2 — a^bi) ki-, + {ciihs — a^bi) kig + 



(^2^3 — asbz) k23. (16) 



axA = Axa = (aiJos + «2^'l3i + aa^is) kiss. (17) 



When the 2-vector in (17) is expressed as a product of two 1- vectors, 

 bxc, that equation becomes 



ax(bxc) = (axb)xc = axbxc = lai a^ a^ kjas. (18) 



\hi h.2 I's 



\Ci C2 C3 



and ax(axb) = 0. (18a) 



"We thus see that axb represents the parallelogram determined by a 

 and b, and axbxc the parallelopiped determined by a, b and c. 



It is important at this point to rewrite equation (13) using bxc in 

 place of A ; expanding and rearranging the terms gives 



a (bxc) ~ abxc 11 = (ac) b — (ab) c. (19) 



These equations (18) and (19) deserve especial attention, for they show 

 the only essential difference between our system and the common 

 system of vector analysis. The two systems give the same result for 

 the outer multiplication of two 1 -vectors and for the inner product of 

 two 1-vectors or two 2-vectors. But the meanings of the outer and 

 inner products of a and bxc are just reversed in the two systems. 



Finally we find from our rules for unit vectors the outer product of 

 two 2-vectors, 



AxB = 0. (20) 



By our general principle the outer product must in this case have the 

 order 2 + 2, and a 4- vector cannot exist in three-dimensional space. ^^ 



^^ The rules here given are somewhat reckindant. For example, the dis- 

 tributive law, and axa = 0, alone suffice to prove axb = — bxa, for 



(a + b) X {a + b) = = axa + bxb + axb -I- bxa = axb + bxa. 

 See Grassmann, Ausdehnungslehre von ISli, p. 87. j 



^^ The parentheses may be removed simply because (ab) X C has no meaning. 



^^ In ordinary vector analysis a meaning is given to the outer pro(Uiot of 

 (axb) and (cxd). It represents a vector determined by the line of intersection 

 of the surface axb and Cxd. We have seen that in n-dimensional space an 

 n-vector has some properties of a scalar or vector of the order n-n. So we may 

 modify our rules of multiplication so that the product of a p-vector and a 



