VECTOR ANALYSIS. 85 



where x lt x 2 , y l , y z , z lt z 2 are scalars. Substituting these values in 



xi + yj+zk, 



we obtain (x l + ix z )i + (y l + iy z )j + (i 



or, if we set p i = x l i + yj + ^, 



we obtain 



We shall call this a bivector, a term which will include a vector as a 

 particular case. When we wish to express a bivector by a single 

 letter, we shall use the small German letters. Thus we may write 



An important case is that in which p l and p 2 have the same direction. 

 The bivector may then be expressed in the form (a+ib)p, in which 

 the vector factor, if we choose, may be a unit vector. In this case, we 

 may say that the bivector has a real direction. In fact, if we express 

 the bivector in the form 



(x l + ix z ) i+(y l + iy 2 )j + (z 1 + iz 2 ) & 



the ratios of the coefficients of i, j, and k, which determine the direc- 

 tion cosines of the vector, will in this case be real. 



2. The consideration that operations upon bivectors may be regarded 

 as operations upon their biscalar x-, y- and z-components is sufficient 

 to show the possibility of a bivector analysis and to indicate what its 

 rules must be. But this point of view does not afford the most simple 

 conception of the operations which we have to perform upon bivectors. 

 It is desirable that the definitions of the fundamental operations should 

 be independent of such extraneous considerations as any system of 

 axes. 



The various signs of our analysis, when applied to bivectors, may 

 therefore be defined as follows, viz., 



The bivector equation 



implies the two vector equations 



/ // i / // 

 jj. = fjL , and v = v . 



4 



r "1 

 /./x J. 



* (a + ib) [n + iv~\ = a/jt.-bv + i[av + bp]. 

 [n + u>](a + ib)=fM-vb + i[fjib + va]. 

 Therefore the position of the scalar factor is indifferent. [MS. note by author.] 



