Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/80

into which the dyadic may be expanded. We shall call it the determinant of the dyadic, and shall denote it by the notation

$\left\vert \Phi \right\vert$

when the dyadic is expressed by a single letter.

If a dyadic is incomplete, its determinant is zero, and conversely.

The determinant of the product of any number of dyadics is equal to the product of their determinants. The determinant of the reciprocal of a dyadic is the reciprocal of the determinant of that dyadic. The determinants of a dyadic and its conjugate are equal.

The relation of the surfaces $\sigma '$ and $\sigma$ may be expressed by the equation

$\sigma '=\left\vert \Phi \right\vert \Phi _{\text{C}}^{-1}.\sigma .$[1]

141. Let us now consider the different cases of rotation and strain as determined by the nature of the dyadic $\Phi .$

If $\Phi$ is reducible to the form

$i'i+j'j+k'k,$

$i,j,k,i',j',k'$ being normal systems of unit vectors (see No. 11), the body will suffer no change of form. For if

$\rho =xi+yj+zk,$

we shall have

$\rho '=xi'+yj'+zk'.$

Conversely, if the body suffers no change of form, the operating dyadic is reducible to the above form. In such cases, it appears from simple geometrical considerations that the displacement of the body may be produced by a rotation about a certain axis. A dyadic reducible to the form

$i'i+j'j+k'k$

may therefore be called a versor.

142. The conjugate operator evidently produces the reverse rotation. A versor, therefore, is the reciprocal of its conjugate.

Conversely, if a dyadic is the reciprocal of its conjugate, it is either a versor, or a versor multiplied by -1. For the dyadic may be expressed in the form

$\alpha i+\beta j+\gamma k.$

Its conjugate will be

$i\alpha +j\beta +k\gamma .$

If these are reciprocals, we have

$\{\alpha i+\beta j\gamma k\}.\{i\alpha +j\beta +k\gamma \}=\alpha \alpha +\beta \beta +\gamma \gamma ={\text{I}}.$

But this relation cannot subsist unless $\alpha ,\beta ,\gamma$ are reciprocals to themselves, i.e., unless they are mutually perpendicular unit-vectors. Therefore, they either are a normal system of unit-vectors, or will become such if their directions are reversed. Therefore, one of the dyadics

$\alpha i+\beta j+\gamma k$⁠and⁠$-\alpha i-\beta j-\gamma k$

is a versor.

1. [See note on p. 90.]