174 PROCEEDINGS OF THE AMERICAN ACADEMY. 



forms of multiplication are completely defined by the distributive law, 

 and by the rules already given for the transposition of subscripts, and 

 for inner and outer multiplication among the unit vectors. We may 

 therefore write at once a large number of equations, of which some of 

 the more important are the following, 



AXB = BXA = (xll2/?34 4- ^13^24 + ^14^23 ^A.^Bu + 



A^,B,, + .434i?12)kl234. (37) 



ax A = Axa = (^1^23 + «2^3i + as^ 12)^123 + (ai^24 + «2-^4i + 



«4^12)kl24 + (ai^34 + «3^41 + «4^ 13)^134 + («2^34 + «'3^42 + 



«4^ 23)^034. (38) 



axb = {((Jj2 - <a'2^i)ki2 + (^1^3 — «3^i)ki3 + . . (39) 



axa = 0. (40) 



Here also axb evidently represents the parallelogram determined by 

 a and b, so axbxc will represent a parallelopiped, and axbxcxd a par- 

 allel four-dimensional figure. It is very important to observe that all 

 of our four-dimensional vector equations may be given simple geo- 

 metrical definitions, and retain complete validity whatever set of coor- 

 dinate axes be arbitrarily chosen. 



Some of the more important inner products are the following : 



ab — ba = aibi -f c/.^o + ffah + aib^. (41) 



AB = BA = ^12^12 + AuBis + AuBu + A.^Bo^ -f A.^B.i + 



A,JJ,,. (42) 



aA = Aa = (^lio^s + -^li3«3 + Au(h)ki -f (Audi + A.^s^s + 



^24rt4)k2 + . . . (43) 



(axb) (cxd) = (ac) (bd) - (bo) (ad). (44) 



a (bxc) = (ac) b — (ab) c. (45) 



This is a 1 -vector lying in the plane bxc and perpendicular to the 

 projection of a thereon. So a (bxcxd) is a 2-vector lying in the 

 3-space bxcxd and perpendicular to tlie projection of a on that 3-space ; 

 (axb) (cxdxe) is a 1- vector in the 3-space cxdxe and perpendicular to 

 the projection of axb thereon. 



The inner product of any vector with unit pseudo-scalar, ki234, is an- 

 other vector of the same magnitude which may be called its comple- 

 ment. The complement of a scalar is a pseudo-scalar, and vice versa. 

 The complement of a 1 -vector is a 3-vector normal to it, and vice versa. 



