Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/100

 The constants $$\alpha$$ and $$\beta$$ are to be determined by 189. It will appear, on reference to Nos. 156–157, that every complete dyadic may be expressed in one of three forms, viz., as a square, as a square with the negative sign, or as a difference of squares of two dyadics of which both the direct products are equal to zero. It follows that every equation of the form where $$\Theta$$ is any constant and complete dyadic, may be integrated by the preceding formulæ.

1. A vector is determined by three algebraic quantities. It often occurs that the solution of the equations by which these are to be detenmined gives imaginary values, i.e., instead of scalars we obtain biscalars, or expressions of the form $$a + \iota \, b,$$ where $$a$$ and $$b$$ are scalars, and $$\iota = \sqrt{-1}.$$ It is most simple, and always allowable, to consider the vector as determined by its components parallel to a normal system of axes. In other words, a vector may be represented in the form Now if the vector is required to satisfy certain conditions, the solution of the equations which determine the values of $$x, y,$$ and $$z,$$ in the most general caae, will give results of the form