Page:Scientific Papers of Josiah Willard Gibbs.djvu/222

But if the strain varies, we may consider $\epsilon _{V'}$ as a function of $\eta _{V'}$ and the nine quantities in (354), and may write

$\delta \epsilon _{V'}=t\delta \eta _{V'}+X_{X'}\delta {\frac {dx}{dx'}}+X_{Y'}\delta {\frac {dx}{dy'}}+X_{Z'}\delta {\frac {dx}{dz'}}+Y_{X'}\delta {\frac {dy}{dx'}}+Y_{Y'}\delta {\frac {dy}{dy'}}+Y_{Z'}\delta {\frac {dy}{dz'}}+Z_{X'}\delta {\frac {dz}{dx'}}+Z_{Y'}\delta {\frac {dz}{dy'}}+Z_{Z'}\delta {\frac {dz}{dz'}},$

(355)

where $X_{X'},...Z_{Z'}$ denote the differential coefficients of $\epsilon _{V'}$ taken with respect to ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}\cdot$ The physical signification of these quantities

will be apparent, if we apply the formula to an element which in the state of reference is a right parallelepiped having the edges $dx',dy',dz',$, and suppose that in the strained state the face in which $x'$ has the smaller constant value remains fixed, while the opposite face is moved parallel to the axis of $X$. If we also suppose no heat to be imparted to the element, we shall have, on multiplying by $dx'dy'dz'$,

$\delta \epsilon _{V'}dx'dy'dz'=X_{X'}\delta {\frac {dx}{dx'}}dx'dy'dz'.$

Now the first member of this equation evidently represents the work done upon the element by the surrounding elements; the second member must therefore have the same value. Since we must regard the forces acting on opposite faces of the elementary parallelepiped as equal and opposite, the whole work done will be zero except for the face which moves parallel to $X$. And since $\delta {\frac {dx}{dx'}}dx'$ represents the distance moved by this face, $X_{X'}dy'dz'$ must be equal to the component parallel to $X$ of the force acting upon this face. In general, therefore, if by the positive side of a surface for which $x'$ is constant we understand the side on which $x'$ has the greater value, we may say that $X_{X'}$ denotes the component parallel to $X$ of the force exerted by the matter on the positive side of a surface for which $x'$ is constant upon the matter on the negative side of that surface per unit of the surface measured in the state of reference. The same may be said, mutatis mutandis, of the other symbols of the same type.

It will be convenient to use $\textstyle \sum$ and $\textstyle \sum '$ to denote summation with respect to quantities relating to the axes $X,Y,Z$, and to the axes $X',Y',Z'$, respectively. With this understanding we may write

$\delta \epsilon _{V'}=t\delta \eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)\cdot$

(356)

This is the complete value of the variation of $\epsilon _{V'}$ for a given element of the solid. If we multiply by $dx'dy'dz'$, and take the integral for