Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/85

Def.—The displacement represented by the equation

$\rho '=-\rho$

is called inversion. The most general case of a homogeneous strain may therefore be produced by a pure strain and a rotation with or without inversion.

150. If

	$\Phi$	$=ai'i+bj'j+ck'k,$
	$\Phi .\Phi _{\text{C}}$	$=a^{2}i'i'+b^{2}j'j'+c^{2}k'k',$
and	$\Phi _{\text{C}}.\Phi$	$=a^{2}ii+b^{2}jj+c^{2}kk.$

The general problem of the determination of the principal ratios and axes of strain for a given dyadic may thus be reduced to the case of a right tensor.

151. Def.—The effect of a prefactor of the form

$a\alpha \alpha '+b\beta \beta '+c\gamma \gamma ',$

where $a,b,c$ are positive or negative scalars, $\alpha ,\beta ,\gamma$ non-complanar vectors, and $\alpha ',\beta ',\gamma '$ their reciprocals, is to change $\alpha$ into $a\alpha ,\beta$ into $b\beta ,$ and $\gamma$ into $c\gamma .$ As a postfactor, the same dyadic will change $\alpha '$ into $a\alpha ',\beta '$ into $b\beta ',$ and $\gamma$ into $c\gamma '.$ Dyadics which can be reduced to this form we shall call tonic (Gr. τείνω). The right tensor already described constitutes a particular case, distinguished by perpendicular axes and positive values of the coefficients $a,b,c.$

The value of the dyadic is evidently not affected by substituting vectors of different lengths but the same or opposite directions for $\alpha ,\beta ,\gamma ,$ with the necessaiy changes in the values of $\alpha ',\beta ',\gamma ',$ defined as reciprocals of $\alpha ,\beta ,\gamma .$ But, except this change, if $a,b,c$ are unequal, the dyadic can be expressed only in one way in the above form. If, however, two of these coefficients are equal, say $\alpha$ and $\beta ,$ any two non-collinear vectors in the $\alpha {\text{-}}\beta$ plane may be substituted for $\alpha$ and $\beta ,$ or, if the three coefficients are equal, any three non-complanar vectors may be substituted for $\alpha ,\beta ,\gamma .$

152. Tonics having the same axes (determined by the directions of $\alpha ,\beta ,\gamma$) are homologous, and their multiplication is effected by multiplying their coefficients. Thus,

${\begin{aligned}\{a_{1}\alpha \alpha '+b_{1}\beta \beta '+c_{1}\gamma \gamma '\}.\{a_{2}\alpha \alpha '+b_{2}\beta \beta '+c_{2}\gamma &\gamma '\}\\&=\{a_{1}a_{2}\alpha \alpha '+b_{1}b_{2}\beta \beta '+c_{1}c_{2}\gamma \gamma '\}.\end{aligned}}$

Hence, division of such dyadics is effected by division of their coefficients. A tonic of which the three coefficients $a,b,c$ are unequal, is homologous only with such dyadics as can be obtained by varying the coefficients.

153. The effect of a prefactor of the form

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

or

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