Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/103

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

${\mathfrak {a}}=m{\mathfrak {b}},$

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 $\cos q+\iota \sin q.$ The effect of multiplying a bivector by a positive scalar is obvious. To understand the effect of a multiplier of the form $\cos q+\iota \sin q$ upon a bivector $\mu +\iota \nu ,$ let us set

$\mu '+\iota \nu '=(\cos q+\iota \sin q)[\mu +\iota \nu ].$

We have then

${\begin{aligned}\mu '&=\cos q\mu -\sin q\nu ,\\\nu '&=\cos q\nu +\sin q\mu .\end{aligned}}$

Now if $\mu$ and $\nu$ 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 $\nu$ to $\mu .$ In any case we may regard $\mu$ and $\nu$ as the projections (by parallel lines) of two perpendicular vectors of the same length. The two last equations show that $\mu '$ and $\nu '$ will be the projections of the vectors obtained by the rotation of these perpendicular vectors in their plane through the angle $q.$ Hence, if we construct an ellipse of which $\mu$ and $\nu$ are conjugate semi-diameters, $\mu '$ and $\nu '$ will be another pair of conjugate semi-diameters, and the sectors between $\mu$ and $\mu '$ and between $\nu$ and $\nu ',$ will each be to the whole area of the ellipse as $q$ to $2\pi ,$ the sector between $\nu$ and $\nu '$ lying on the same side of $\nu$ and $\mu ,$ and that between $\mu$ and $\mu '$ lying on the same side of $\mu$ as $-\nu .$

It follows that any bivector $\mu +\iota \nu$ may be put in the form

$(\cos q+\iota \sin q)[\alpha +\iota \beta ],$

in which $\alpha$ and $\beta$ are at right angles, being the semi-axes of the ellipse of which $\mu$ and $\nu$ 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 imaginaiy 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 identity of the angular directions mentioned above.