Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/101

Rh where $$x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2}$$ are scalars. Substituiing these values in we obtain  or, if we set  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 $$\rho_{1}$$ and $$\rho_{2}$$ have the same direction. The bivector may then be expressed in the form $$(a + \iota b)\rho,$$ 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 the ratios of the coefficients of $$i, j,$$ and $$k,$$ which determine the direction 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