Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/90

This dyadic is homologous with snch as are obtained by varying the values of $a,b,c,$ and only such, unless $b=0.$

It is always possible to take three mutually perpendicular vectors for $\alpha ,\beta ,$ and $\gamma ;$ or, if it be preferred, to take such values for these vectors as shall make the term containing $c$ vanish.

158. The dyadics described in the two last paragraphs may be called shearing dyadics.

The criterion of a shearer is

$\{\Phi -{\text{I}}\}^{3}=0,$⁠$\Phi -{\text{I}}\neq 0.$

The criterion of a simple shearer is

$\{\Phi -{\text{I}}\}^{2}=0,$⁠$\Phi -{\text{I}}\neq 0.$

The criterion of a complex shearer is

$\{\Phi -{\text{I}}\}^{2}=0,$⁠$\{\Phi -{\text{I}}\}^{2}\neq 0.$

Note.—If a dyadic $\Phi$ is a linear function of a vector $p$ (the term linear being used in the same sense as in No. 105), we may represent the relation by an equation of the form

$\Phi =\alpha \beta \,\gamma .\rho +e\zeta \,\eta .\rho +{\text{etc.,}}$

or

$\Phi =\{\alpha \beta \gamma +e\zeta \eta +{\text{etc.,}}\}.\rho ,$

where the expression in the braces may be called a triadic polynomial, and a single term $\alpha \beta \gamma$ a triad, or the indeterminate product of the three vectors $\alpha ,\beta ,\gamma .$ We are thus led successively to the consideration of higher orders of indeterminate products of vectors, triads, tetrads, etc., in general polyads, and of polynomials consisting of such terms, triadics, tetradics, etc., in general polyadics. But the development of the subject in this direction lies beyond our present purpose.

It may sometimes be convenient to use notations like

${\frac {\lambda ,\mu ,\nu }{\vert \alpha ,\beta ,\gamma }}$⁠and⁠${\frac {\lambda ,\mu ,\nu }{\alpha ,\beta ,\gamma \vert }}$

to represent the conjugate dyadics which, the first as prefactor, and the second as postfactor, change $\alpha ,\beta ,\gamma$ into $\lambda ,\mu ,\nu ,$ respectively. In the notations of the preceding chapter these would be written

$\lambda \alpha '+\mu \beta '+\nu \gamma '$⁠and⁠$\alpha '\lambda +\beta '\mu +\gamma '\nu$

respectively, $\alpha ',\beta ',\gamma '$ denoting the reciprocals of $\alpha ,\beta ,\gamma .$ If $\tau$ is a linear function of $\rho ,$ the dyadics which as prefactor and postfactor change $\rho$ into $\tau$ may be written respectively

${\frac {\tau }{\vert \rho }}$⁠and⁠${\frac {\tau }{\rho \vert }}\cdot$

If $\tau$ is any function of $\rho ,$ the dyadics which as prefactor and postfactor change $d\rho$ into $d\tau$ may be written respectively

${\frac {d\tau }{\vert d\rho }}$⁠and⁠${\frac {d\tau }{d\rho \vert }}\cdot$

In the notation of the following chapter the second of these (when $\rho$ denotes a position-vector) would be written $\nabla \tau .$ The triadic which as prefactor changes $d\rho$ into ${\frac {d\tau }{\vert d\rho }}$ may be written ${\frac {d^{2}\tau }{\vert d\rho ^{2}}},$ and that which as postfactor changes $d\rho$ into ${\frac {d\tau }{d\rho \vert }}$ may be written ${\frac {d^{2}\tau }{d\rho ^{2}\vert }}\cdot$ The latter would be written $\nabla \nabla \tau$ in the notations of the following chapter.