of Four-Dimensional Vector Analysis. 



393 



§ 3. The laws of multiplication lead to certain relations 

 between products of the i's of which the following are 

 illustrative : 



l 2 l 2 ll ~~ 





} 



(3-1) 



A change of position of any vector by an odd number of 

 places in any product changes the sign, unless the product 

 is scalar, or contains u scalar part. 



Thus 



afiy = — /3a 7 = ffyot . 



Products of the type i^i 2 i z are not reducible. 



In multiplying three vectors it is convenient to regard 

 the product as composed of reducible and irreducible parts. 

 lnus 



P\PiPz = ViP\P2p* + 'ViPip2Pa', . . . (3*2) 



^ T 3pip2Pz contains terms in i 2 i s i^ 



Ulil 



4*1? 



etc., and Y 1 p 1 p 2 ps 



contains terms in i x i 2 i s and i± which are the result of such 

 reduction as is illustrated in (3'1). 



If p l and p 2 have the values in (2'3) and 



p 3 = a# x + b z i 2 + c 3 i 3 + d 3 i± 



the expansion of the product ^zp\p 2 p?, gives : 



^2pip2pn=hkH(biC 2 dz) + Wi( c A«2) 



(3-3) 



wnere 



i(AiB,C)e 



A 2 -t>2 ^-'2 



A 3 B 3 C 3 



The determinants are recognized as three-dimensional 

 volumes. They are the components of the directed volume 

 Y 3 Pjp 2 p3 on the various sets of three dimensions that may be 

 chosen from the four. 



The coefficients of these irreducible products are minors of 

 the determinant (a-fi 2 Czd^). and in writing the components 

 V x V 2 V 3 V 4 we shall adopt the rule of signs : 



V 1 = (6 1 c 2 ^ 3 ), 



Y 2 =~(c l d 2 a 3 ), 

 Y 4 =-(a 1 h 2 r 3 ). 



y 3 =(<w> 3 ), 



