LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 171 



Tlie inner product of (axb) and (cxd) may be obtained from the 

 preceding eijuations, and is the same as in ordinary vector analysis, 



(axb) (cxd) = (ac) (bd) — (be) (ad). (21) 



The differen^al operator v (" del") we may define in the usual way,!^ 

 namely, 



V=ki 4-k2T— +k3--. (22) 



o.ri du-.i dxz 



Since the scalar operator — , when applied to a single variable, can be 



treated as an algebraic quantity, the operator v may be treated for- 

 mally as a 1 -vector, and we may derive a number of important equa- 

 tions by substituting V for a or b in the preceding equations. Thus 

 from (8) we obtain from the scalar <p the function known as gradient 

 of<^ 



V0 = k,|^ + k.^^^-l-k3|-^. (23) 



dxi (Jj'2 dxz ' 



Combined with a 1 -vector by inner and outer multiplication we ob- 

 tain by equations (10) and (16), the functions known as the divergence 

 and curl, respectively. 



va = — + -^ + -^. (24) 



bi\ dxo, dx3 ^ ^ 



Evidently V0 is a 1-yector, va a scalar, and vxa a 2- vector. 

 By equations (13) and (17) we may write expressions for vA (a 

 1 -vector), and vxA (a 3- vector, or pseudo scalar). 



g-vector, when p + q> n, is a vector of the order p + q — n. Such a product, 

 which Grassmann calls "regressive," is formed according to a new set of rules 

 and may best be regarded as a new type of product entirely distinct from the 

 regular or "progressive" outer product. It is possible in the system here 

 described to avoid the introduction of this new kind of product. Thus 



((axb) ki23) X ( (cxd kjog) = eki^z), 



where e is the 1-vector obtained in the ordinary vector analj^sis as the outer 

 product of (axb) and (cxd). 



^^ Like other vector quantities and operators V may be simply defined 

 without reference to coordinates. See, for example, Wilson, Bull. Amcr. 

 Math. See. (2) 16, 415 (1910). 



