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

The constants $\alpha$ and $\beta$ are to be determined by

${\begin{aligned}\rho _{t=0}&=\alpha ,\\\left[{\frac {d\rho }{dt}}\right]_{t=0}&=\{\Gamma +\Lambda \}.\beta ,\end{aligned}}$

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

${\frac {d^{2}\rho }{dt^{2}}}=\Theta .\rho$

where $\Theta$ is any constant and complete dyadic, may be integrated by the preceding formulæ.

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note on bivector analysis.^[1]

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

$xi+yj+zk.$

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

${\begin{aligned}x&=x_{1}+\iota x_{2},\\y&=y_{1}+\iota y_{2},\\z&=z_{1}+\iota z_{2},\end{aligned}}$

1. Thus far, in accordanoe with the purpoee expressed in the footnote on page 17, we have considered only real values of soalars and vectors. The object of this limitation has been to present the subject in the most elementary manner. The limitation is however often inconvenient, and does not allow the most symmetrical and complete development of the subject in many important directions. Thus in Chapter V, and the latter part of Chapter III, the exclusion of imaginary values has involved a considerable sacrifice of simplicity both in the enunciation of theorems and in their demonstration. The student will find an interesting and profitable exercise in working over this part of the subject with the aid of imaginary values, especially in the discussion of the imaginary roots of the cubic equation on page 71, and in the use of the formula

   $e^{\iota \Phi }=\cos \Phi +\iota \sin \Phi$

   in developing the properties of the sines, cosines, and exponentials of dyadics.