VECTOR ANALYSIS. 87 



where n is a scalar, we say that the vectors are parallel. Analogy 

 leads us to call two bivectors parallel, when there subsists between 



them a relation of the form 



a = rab, 



where m (in the most general case) is a biscalar. 



To aid us in comprehending the geometrical signification of this 

 relation, we may regard the biscalar as consisting of two factors, one 

 of which is a positive scalar (the modulus of the biscalar), and the 

 other may be put in the form cos q + 1 sin q. The effect of multiplying 

 a bivector by a positive scalar is obvious. To understand the effect of 

 a multiplier of the form cosq + t sing upon a bivector p + tv, let us set 



fi'+tv (cos g + 1 sin q) [/A + II/]. 



We have then 



fj! = cos q JUL sin q v, 



v cos q v -f sin q JUL. 



Now if JUL and v are of the same magnitude and at right angles, the 

 effect of the multiplication is evidently to rotate these vectors in 

 their plane an angular distance q, which is to be measured in the 

 direction from v to JUL. In any case we may regard JUL and v as the pro- 

 jections (by parallel lines) of two perpendicular vectors of the same 

 length. The two last equations show that fj! and v will be the 

 projections of the vectors obtained by the rotation of these perpendi- 

 cular vectors in their plane through the angle q. Hence, if we 

 construct an ellipse of which JUL and v are conjugate semi-diameters, JUL' 

 and v will be another pair of conjugate semi-diameters, and the sectors 

 between JUL and //' and between v and v ', will each be to the whole area 

 of the ellipse as q to 27r, the sector between v and v lying on the same 

 side of v and JUL, and that between JUL and JUL' lying on the same side 

 of ju as v. 



It follows that any bivector /* + / may be put in the form 



(cos q + 1 sin q) [a 4- */3], 



in which a and /3 are at right angles, being the semi-axes of the ellipse 

 of which JUL and v are conjugate semi-diameters. This ellipse we may 

 call the directional ellipse of the bivector. In the case of a real 

 vector, or of a vector having a real direction, it reduces to a straight 

 line. In any other case, the angular direction from the imaginary to 

 the real part of the bivector is to be regarded as positive in the ellipse, 

 and the specification of the ellipse must be considered incomplete 

 without the indication of this direction. 



Parallelism of bivectors, then, signifies the similarity and similar 

 position of their directional ellipses. Similar position includes iden- 

 tity of the angular directions mentioned above. 



