Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/89

Therefore, since

$\{a\alpha \alpha '+p\Psi \}.\alpha =a\alpha =\Phi .\alpha ,$ $\{a\alpha \alpha '+p\Psi \}.\beta =p\beta _{1}=\Phi .\beta ,$ $\{a\alpha \alpha '+p\Psi \}.\beta _{-1}=p\beta =\Phi .\beta _{-1},$

it follows (by No. 108) that

$\Phi =a\alpha \alpha '+p\Psi =a\alpha \alpha '+p\,\cos q\{\beta \beta '+\gamma \gamma '\}+p\,\sin q\{\gamma \beta '-\beta \gamma '\}.$

156. It will be sufficient to indicate (without demonstration) the forms of dyadics which belong to the particular cases which have been passed over in the preceding paragraph, so far as they present any notable peculiarities.

If $n=\pm 1,$ (page 72), the dyadic may be reduced to the form

$a\alpha \alpha '+b\{\beta \beta '+\gamma \gamma '\}+bc\beta \gamma ',$

where $\alpha ,\beta ,\gamma$ are three non-complanar vectors, $\alpha ',\beta ',\gamma '$ their reciprocals, and $a,b,c$ positive or negative scalars. The effect of this as an operator, will be evident if we resolve it into the three homologous factors

${\begin{aligned}a\alpha \alpha '&+\beta \beta '+\gamma \gamma ',\\\alpha \alpha '&+b\{\beta \beta '+\gamma \gamma '\},\\\alpha \alpha '&+\beta \beta '+\gamma \gamma '+c\beta \gamma '.\end{aligned}}$

The displacement due to the last factor may be called a simple shear. It consists (when the dyadic is used as pref actor) of a motion parallel to $\beta ,$ and proportioned to the distance from the $\alpha {\text{-}}\beta$ plane. This factor may be called a shearer.

This dyadic is homologous with such as are obtained by varying the values of $a,b,c,$ and only with such, when the values of $a$ and $b$ are different, and that of $c$ other than zero.

157. If the planar $\Phi -a{\text{I}}$ (page 71) has perpendicular planes, there may be another value of $a,$ of the same sign as $\left\vert \Phi \right\vert ,$ which will give a planar which has not perpendicular planes. When this is not the case, the dyadic may always be reduced to the form

$a\{\alpha \alpha '+\beta \beta '+\gamma \gamma '\}+ab\{\alpha \beta '+\beta \gamma '\}+ae\alpha \gamma ',$

where $\alpha ,\beta ,\gamma$ are three non-complanar vectors, $\alpha ',\beta ',\gamma ',$ their reciprocals, and $a,b,c,$ positive or negative scalars. This may be resolved into the homologous factors

$a{\text{I}}$and${\text{I}}+b\{\alpha \beta '+\beta \gamma '\}+c\alpha \gamma '.$

The displacement due to the last factor may be called a complex shear. It consists (when the dyadic is used as prefactor) of a motion parallel to a which is proportional to the distance from the $\alpha {\text{-}}\gamma$ plane, together with a motion parallel to $b\beta +c\alpha$ which is proportional to the distance from the $\alpha {\text{-}}\beta$ plane. This factor may be called a complex shearer.