Page:Scientific Papers of Josiah Willard Gibbs.djvu/229

The projection of $L$ on the Y-Z plane will be a triangle, the angular points of which are determined by the co-ordinates

$y,z;$ $y+{\frac {dy}{dy'}}dy',$ $z+{\frac {dz}{dy'}}dy';$ $y+{\frac {dy}{dz'}}dz',$ $z+{\frac {dz}{dz'}}dz';$

the area of such a triangle is

${\tfrac {1}{2}}\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)dy'dz',$

or, since ${\tfrac {1}{2}}dy'dz'$ represents the area of $L'$,

$\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)\alpha 'Ds'.$

(That this expression has the proper sign will appear if we suppose for the moment that the strain vanishes.) The areas of the projections of $M$ and $N$ upon the same plane will be obtained by changing $y',z'$ and $\alpha '$ in this expression into $z',x'$, and $\beta '$, and into $x',y',$ and $\gamma '$. The sum of the three expressions may be substituted for a $Ds$ in (381).

We shall hereafter use $\textstyle \sum '$ to denote the sum of the three terms obtained by rotary substitutions of quantities relating to the axes $X',Y',Z'$ (i.e., by changing $x',y',z'$ into $y',z',x',$ and into $z',x',y',$ with similar changes in regard to $\alpha ',\beta ',\gamma ',$ and other quantities relating to these axes), and $\textstyle \sum$ to denote the sum of the three terms obtained by similar rotary changes of quantities relating to the axes $X,Y,Z$. This is only an extension of our previous use of these symbols.

With this understanding, equations (381) may be reduced to the form

$\textstyle \sum '\displaystyle (\alpha 'X_{X'})+p\textstyle \sum '\displaystyle \left\lbrace \alpha '\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)\right\rbrace =0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ (382)
etc.

The formula (372) expresses the additional condition of equilibrium which relates to the dissolving of the solid, or its growth without discontinuity. If the solid consists entirely of substances which are actual components of the fluid, and there are no passive resistances which impede the formation or dissolving of the solid, $\delta N'$ may have either positive or negative values, and we must have

$\epsilon _{V'}-t\eta _{V'}-pv_{V'}=\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}').$

(383)

But if some of the components of the solid are only possible components (see page 64) of the fluid, $\delta N'$ is incapable of positive values, as the quantity of the solid cannot be increased, and it is sufficient for equilibrium that

$\epsilon _{V'}-t\eta _{V'}-pv_{V'}\leqq \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}').$

(384)

To express- condition (383) in a form independent of the state of reference, we may use $\epsilon _{V},\eta _{V},\Gamma _{1}$, etc., to denote the densities of