# Scientific Papers of Josiah Willard Gibbs, Volume 1/Chapter IIIb

Scientific Papers of Josiah Willard Gibbs, Volume 1 by Josiah Willard Gibbs
On the Equilibrium of Heterogeneous Substances, Part 2

[1]The Conditions of Internal and External Equilibrium for Solids in contact with Fluids with regard to all possible States of Strain of the Solids.

In treating of the physical properties of a solid, it is necessary to consider its state of strain. A body is said to be strained when the relative position of its parts is altered, and by its state of strain is meant its state in respect to the relative position of its parts. We have hitherto considered the equilibrium of solids only in the case in which their state of strain is determined by pressures having the same values in all directions about any point. Let us now consider the subject without this limitation.

If ${\displaystyle x',y',z'}$ are the rectangular co-ordinates of a point of a solid body in any completely determined state of strain, which we shall call the state of reference, and ${\displaystyle x,y,z}$, the rectangular co-ordinates of the same point of the body in the state in which its properties are the subject of discussion, we may regard ${\displaystyle x,y,z}$ as functions of ${\displaystyle x',y',z'}$, the form of the functions determining the second state of strain. For brevity, we may sometimes distinguish the variable state, to which ${\displaystyle x,y,z}$ relate, and the constant state (state of reference) to which ${\displaystyle x',y',z'}$ relate, as the strained and unstrained states; but it must be remembered that these terms have reference merely to the change of form or strain determined by the functions which express the relations of ${\displaystyle x,y,z}$ and ${\displaystyle x',y',z'}$, and do not imply any particular physical properties in either of the two states, nor prevent their possible coincidence. The axes to which the co-ordinates ${\displaystyle x,y,z}$ and ${\displaystyle x',y',z'}$ relate will be distinguished as the axes of ${\displaystyle X,Y,Z}$ and ${\displaystyle X',Y',Z'}$. It is not necessary, nor always convenient, to regard these systems of axes as identical, but they should be similar, i.e., capable of superposition.

The state of strain of any element of the body is determined by the values of the differential coefficients of ${\displaystyle x,y,}$ and ${\displaystyle z}$ with respect to ${\displaystyle x',y',}$, and ${\displaystyle z'}$; for changes in the values of ${\displaystyle x,y,z}$, when the differential coefficients remain the same, only cause motions of translation of the body. When the differential coefficients of the first order do not vary sensibly except for distances greater than the radius of sensible molecular action, we may regard them as completely determining the state of strain of any element. There are nine of these differential coefficients, viz.,

 ${\displaystyle {\frac {dx}{dx}},}$⁠ ${\displaystyle {\frac {dx}{dy}},}$⁠ ${\displaystyle {\frac {dx}{dz}},}$⁠ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (354) ${\displaystyle {\frac {dy}{dx}},}$⁠ ${\displaystyle {\frac {dy}{dy}},}$⁠ ${\displaystyle {\frac {dy}{dz}},}$⁠ ${\displaystyle {\frac {dz}{dx}},}$⁠ ${\displaystyle {\frac {dz}{dy}},}$⁠ ${\displaystyle {\frac {dz}{dz}}\cdot }$⁠

It will be observed that these quantities determine the orientation of the element as well as its strain, and both these particulars must be given in order to determine the nine differential coefficients. Therefore, since the orientation is capable of three independent variations, which do not affect the strain, the strain of the element, considered without regard to directions in space, must be capable of six independent variations.

The physical state of any given element of a solid in any unvarying state of strain is capable of one variation, which is produced by addition or subtraction of heat. If we write ${\displaystyle \epsilon _{V}}$ and ${\displaystyle \eta _{V}}$ for the energy and entropy of the element divided by its volume in the state of reference, we shall have for any constant state of strain

 ${\displaystyle \delta \epsilon _{V}=t\delta \eta _{V}.}$
But if the strain varies, we may consider ${\displaystyle \epsilon _{V'}}$ as a function of ${\displaystyle \eta _{V'}}$ and the nine quantities in (354), and may write
 ${\displaystyle \delta \epsilon _{V'}=t\delta \eta _{V'}+X_{X'}\delta {\frac {dx}{dx'}}+X_{Y'}\delta {\frac {dx}{dy'}}+X_{Z'}\delta {\frac {dx}{dz'}}+Y_{X'}\delta {\frac {dy}{dx'}}+Y_{Y'}\delta {\frac {dy}{dy'}}+Y_{Z'}\delta {\frac {dy}{dz'}}+Z_{X'}\delta {\frac {dz}{dx'}}+Z_{Y'}\delta {\frac {dz}{dy'}}+Z_{Z'}\delta {\frac {dz}{dz'}},}$ (355)
where ${\displaystyle X_{X'},...Z_{Z'}}$ denote the differential coefficients of ${\displaystyle \epsilon _{V'}}$ taken with respect to ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}\cdot }$ 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 ${\displaystyle dx',dy',dz',}$, and suppose that in the strained state the face in which ${\displaystyle x'}$ has the smaller constant value remains fixed, while the opposite face is moved parallel to the axis of ${\displaystyle X}$. If we also suppose no heat to be imparted to the element, we shall have, on multiplying by ${\displaystyle dx'dy'dz'}$,

 ${\displaystyle \delta \epsilon _{V'}dx'dy'dz'=X_{X'}\delta {\frac {dx}{dx'}}dx'dy'dz'.}$
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 ${\displaystyle X}$. And since ${\displaystyle \delta {\frac {dx}{dx'}}dx'}$ represents the distance moved by this face, ${\displaystyle X_{X'}dy'dz'}$ must be equal to the component parallel to ${\displaystyle X}$ of the force acting upon this face. In general, therefore, if by the positive side of a surface for which ${\displaystyle x'}$ is constant we understand the side on which ${\displaystyle x'}$ has the greater value, we may say that ${\displaystyle X_{X'}}$ denotes the component parallel to ${\displaystyle X}$ of the force exerted by the matter on the positive side of a surface for which ${\displaystyle x'}$ 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 ${\displaystyle \textstyle \sum }$ and ${\displaystyle \textstyle \sum '}$ to denote summation with respect to quantities relating to the axes ${\displaystyle X,Y,Z}$, and to the axes ${\displaystyle X',Y',Z'}$, respectively. With this understanding we may write

 ${\displaystyle \delta \epsilon _{V'}=t\delta \eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)\cdot }$ (356)
This is the complete value of the variation of ${\displaystyle \epsilon _{V'}}$ for a given element of the solid. If we multiply by ${\displaystyle dx'dy'dz'}$, and take the integral for the whole body, we shall obtain the value of the variation of the total energy of the body, when this is supposed invariable in substance. But if we suppose the body to be increased or diminished in substance at its surface (the increment being continuous in nature and state with the part of the body to which it is joined), to obtain the complete value of the variation of the energy of the body, we must add the integral
 ${\displaystyle \int \epsilon _{V'}\delta N'Ds'}$
in which ${\displaystyle Ds'}$ denotes an element of the surface measured in the state of reference, and ${\displaystyle \delta N'}$ the change in position of this surface (due to the substance added or taken away) measured normally and outward in the state of reference. The complete value of the variation of the intrinsic energy of the solid is therefore
 ${\displaystyle \iiint t\delta \eta _{V'}dx'dy'dz'+\iiint \textstyle \sum \sum ^{\prime }\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)+\int \epsilon _{V'}\delta N'Ds'.}$ (357)
This is entirely independent of any supposition in regard to the homogeneity of the solid.

To obtain the conditions of equilibrium for solid and fluid masses in contact, we should make the variation of the energy of the whole equal to or greater than zero. But since we have already examined the conditions of equilibrium for fluids, we need here only seek the conditions of equilibrium for the interior of a solid mass and for the surfaces where it comes in contact with fluids. For this it will be necessary to consider the variations of the energy of the fluids only so far as they are immediately connected with the changes in the solid. We may suppose the solid with so much of the fluid as is in close proximity to it to be enclosed in a fixed envelop, which is impermeable to matter and to heat, and to which the solid is firmly attached wherever they meet. We may also suppose that in the narrow space or spaces between the solid and the envelop, which are filled with fluid, there is no motion of matter or transmission of heat across any surfaces which can be generated by moving normals to the surface of the solid, since the terms in the condition of equilibrium relating to such processes may be cancelled on account of the internal equilibrium of the fluids. It will be observed that this method is perfectly applicable to the case in which a fluid mass is entirely enclosed in a solid. A detached portion of the envelop will then be necessary to separate the great mass of the fluid from the small portion adjacent to the solid, which alone we have to consider. Now the variation of the energy of the fluid mass will be, by equation (13),

 ${\displaystyle \int ^{F}t\delta D\eta -\int ^{F}p\delta Dv+\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1},}$ (358)
where ${\displaystyle \int ^{F}}$ denotes an integration extending over all the elements of the fluid (within the envelop), and ${\displaystyle \textstyle \sum _{1}}$ denotes a summation with regard to those independently variable components of the fluid of which the solid is composed. Where the solid does not consist of substances which are components, actual or possible (see page 64), of the fluid, this term is of course to be cancelled.

If we wish to take account of gravity, we may suppose that it acts in the negative direction of the axis of ${\displaystyle Z}$. It is evident that the variation of the energy due to gravity for the whole mass considered is simply

 ${\displaystyle \iiint g\Gamma '\delta z\delta x'\delta y'\delta z',}$ (359)
where ${\displaystyle g}$ denotes the force of gravity, and ${\displaystyle \Gamma '}$ the density of the element in the state of reference, and the triple integration, as before, extends throughout the solid.

We have, then, for the general condition of equilibrium,

 ${\displaystyle \iiint t\delta \eta _{V'}dx'dy'dz'+\iiint \textstyle \sum \sum \,'\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)dx'dy'dz'+\iiint g\Gamma '\delta z\delta x'\delta y'\delta z'+\int \epsilon _{V'}\delta N'Ds'+\int ^{F}t\delta D\eta -\int ^{F}p\delta Dv+\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1}\geqq 0.}$ (360)
The equations of condition to which these variations are subject are:

(1) that which expresses the constancy of the total entropy,

 ${\displaystyle \iiint \delta \eta _{V'}\delta x'\delta y'\delta z'+\int \eta _{V'}\delta N'Ds'+\int ^{F}\delta D\eta =0;}$ (361)
(2) that which expresses how the value of ${\displaystyle \delta Dv}$ for any element of the fluid is determined by changes in the solid,
 ${\displaystyle \delta Dv=-(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds-v_{V'}\delta N'Ds',}$ (362)
where ${\displaystyle \alpha ,\beta ,\gamma }$ denote the direction cosines of the normal to the surface of the body in the state to which ${\displaystyle x,y,z}$ relate, ${\displaystyle Ds}$ the element of the surface in this state corresponding to ${\displaystyle Ds'}$ in the state of reference, and ${\displaystyle v_{V'}}$ the volume of an element of the solid divided by its volume in the state of reference;

(3) those which express how the values of ${\displaystyle \delta Dm_{1},\delta Dm_{2}}$, etc. for any element in the fluid are determined by the changes in the solid,

 ${\displaystyle \delta Dm_{1}=-\Gamma _{1}'\delta N'Ds',}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}}$ (363) ${\displaystyle \delta Dm_{2}=-\Gamma _{2}'\delta N'Ds',}$ etc.,

where ${\displaystyle \Gamma _{1}',\Gamma _{2}'}$, etc. denote the separate densities of the several components in the solid in the state of reference.

Now, since the variations of entropy are independent of all the other variations, the condition of equilibrium (360), considered with regard to the equation of condition (361), evidently requires that throughout the whole system

 ${\displaystyle t=const.}$ (364)

We may therefore use (361) to eliminate the fourth and fifth integrals from (360). If we multiply (362) by ${\displaystyle p}$, and take the integrals for the whole surface of the solid and for the fluid in contact with it, we obtain the equation {{MathForm2|(365)|${\displaystyle \int ^{F}p\delta Dv=-\int p(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds-\int pv_{V'}\delta N'Ds',}$ by means of which we may eliminate the sixth integral from (360). If we add equations (363) multiplied respectively by ${\displaystyle \mu _{1},\mu _{2}}$, etc., and take the integrals, we obtain the equation

 ${\displaystyle \textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1}=-\int \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')\delta N'Ds',}$ (366)
by means of which we may eliminate the last integral from (360).

The condition of equilibrium is thus reduced to the form

 ${\displaystyle \iiint \textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)dx'dy'dz'+\iiint g\gamma '\delta zdx'dy'dz'+\int \epsilon _{V'}\delta N'Ds'-\int t\eta _{V'}\delta N'Ds'+\int p(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds+\int pv_{V'}\delta N'Ds'-\int \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')\delta N'Ds'\geqq 0,}$ (367)
in which the variations are independent of the equations of condition, and in which the only quantities relating to the fluids are ${\displaystyle p}$ and ${\displaystyle \mu _{1},\mu _{2}}$, etc.

Now by the ordinary method of the calculus of variations, if we write ${\displaystyle \alpha ',\beta ',\gamma ',}$ for the direction cosines of the normal to the surface of the solid in the state of reference, we have

 ${\displaystyle \iiint X_{X'}\delta {\frac {dx}{dx'}}dx'dy'dz'=\int \alpha 'X_{X'}\delta xDs'-\iiint {\frac {dX_{X'}}{dx'}}\delta xdx'dy'dz',}$ (368)
with similar expressions for the other parts into which the first integral in (367) may be divided. The condition of equilibrium is thus reduced to the form
 ${\displaystyle -\iiint \textstyle \sum \sum '\displaystyle \left({\frac {dX_{X'}}{dx'}}\delta x\right)dx'dy'dz'+\iiint g\gamma '\delta zdx'dy'dz'+\int \textstyle \sum \sum '\displaystyle (\alpha 'X_{X'}\delta x)Ds'+\int p\textstyle \sum \displaystyle (\alpha \delta x)Ds+\int [\epsilon _{V'}-t\eta _{V'}+pv_{V'}-\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')]\delta N'Ds'\geqq 0.}$ (369)
It must be observed that if the solid mass is not continuous throughout in nature and state, the surface-integral in (368), and therefore the first surface-integral in (369), must be taken to apply not only to the external surface of the solid, but also to every surface of discontinuity within it, and that with reference to each of the two masses separated by the surface. To satisfy the condition of equilibrium, as thus understood, it is necessary and sufficient that throughout the solid mass
 ${\displaystyle \textstyle \sum \sum '\displaystyle \left({\frac {dX_{X'}}{dx'}}\delta x\right)-g\Gamma '\delta z=0;}$ (370)
that throughout the surfaces where the solid meets the fluid
 ${\displaystyle Ds'\textstyle \sum \sum '\displaystyle (\alpha 'X_{X'}\delta x)+Dsp\textstyle \sum \displaystyle (\alpha \delta x)=0,}$ (371)
 and⁠${\displaystyle [\epsilon _{V'}-t\eta _{V'}+pv_{V'}-\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')]\delta N'\geqq 0;}$ (372)
and that throughout the internal surfaces of discontinuity where the suffixed numerals distinguish the expressions relating to the masses on opposite sides of a surface of discontinuity.

Equation (370) expresses the mechanical conditions of internal equilibrium for a continuous solid under the influence of gravity. If we expand the first term, and set the coefficients of ${\displaystyle \delta x,\delta y}$, and ${\displaystyle \delta z}$ separately equal to zero, we obtain

 ${\displaystyle {\frac {dX_{X'}}{dx'}}+{\frac {dX_{Y'}}{dy'}}+{\frac {dX_{Z'}}{dz'}}=0,}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (374) ${\displaystyle {\frac {dY_{X'}}{dx'}}+{\frac {dY_{Y'}}{dy'}}+{\frac {dY_{Z'}}{dz'}}=0,}$ ${\displaystyle {\frac {dZ_{X'}}{dx'}}+{\frac {dZ_{Y'}}{dy'}}+{\frac {dZ_{Z'}}{dz'}}=g\Gamma ',}$

The first member of any one of these equations multiplied by ${\displaystyle dx'dy'dz'}$ evidently represents the sum of the components parallel to one of the axes ${\displaystyle X,Y,Z}$ of the forces exerted on the six faces of the element ${\displaystyle dx'dy'dz'}$ by the neighboring elements.

As the state which we have called the state of reference is arbitrary, it may be convenient for some purposes to make it coincide with the state to which ${\displaystyle x,y,z}$ relate, and the axes ${\displaystyle X',Y',Z'}$ with the axes ${\displaystyle X,Y,Z}$. The values of ${\displaystyle X_{X'},...Z_{Z'}}$ on this particular supposition may be represented by the symbols ${\displaystyle X_{X},...Z_{Z}}$. Since

 ${\displaystyle X_{Y'}={\frac {d\epsilon _{V'}}{d{\frac {dx}{dy'}}}},}$⁠and⁠${\displaystyle Y_{X'}={\frac {d\epsilon _{V'}}{d{\frac {dy}{dx'}}}},}$
and since, when the states, ${\displaystyle x,y,z}$ and ${\displaystyle x',y',z'}$ coincide, and the axes ${\displaystyle X,Y,Z,}$ and ${\displaystyle X',Y',Z',d{\frac {dx}{dy'}}}$, and ${\displaystyle d{\frac {dy}{dx'}}}$, represent displacements which differ only by a rotation, we must have
 ${\displaystyle X_{Y}=Y_{X},}$ (375)
and for similar reasons,
 ${\displaystyle Y_{Z}=Z_{Y},}$⁠${\displaystyle Z_{X}=X_{Z}.}$ (376)

The six quantities ${\displaystyle X_{X},Y_{Y},Z_{Z},X_{Y}}$, or ${\displaystyle Y_{X},Y_{Z}}$ or ${\displaystyle Z_{Y}}$, and ${\displaystyle Z_{X}}$ or ${\displaystyle X_{Z}}$ are called the rectangular components of stress, the three first being the longitudinal stresses and the three last the shearing stresses. The mechanical conditions of internal equilibrium for a solid under the influence of gravity may therefore be expressed by the equations

 ${\displaystyle {\frac {dX_{X}}{dx}}+{\frac {dX_{Y}}{dy}}+{\frac {dX_{Z}}{dz}}=0,}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (377) ${\displaystyle {\frac {dY_{X}}{dx}}+{\frac {dY_{Y}}{dy}}+{\frac {dY_{Z}}{dz}}=0,}$ ${\displaystyle {\frac {dZ_{X}}{dx}}+{\frac {dZ_{Y}}{dy}}+{\frac {dZ_{Z}}{dz}}=g\Gamma ,}$

where ${\displaystyle \Gamma }$ denotes the density of the element to which the other symbols relate. Equations (375), (376) are rather to be regarded as expressing necessary relations (when ${\displaystyle X_{X},...Z_{Z}}$ are regarded as internal forces determined by the state of strain of the solid) than as expressing conditions of equilibrium. They will hold true of a solid which is not in equilibrium,—of one, for example, through which vibrations are propagated,—which is not the case with equations (377).

Equation (373) expresses the mechanical conditions of equilibrium for a surface of discontinuity within the solid. If we set the coefficients of ${\displaystyle \delta x,\delta y,\delta z}$, separately equal to zero we obtain

 ${\displaystyle (\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'})_{1}+(\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'})_{2}=0,}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (378) ${\displaystyle (\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'})_{1}+(\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'})_{2}=0,}$ ${\displaystyle (\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'})_{1}+(\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'})_{2}=0.}$

Now when the ${\displaystyle \alpha ',\beta ',\gamma '}$ represent the direction-cosines of the normal in the state of reference on the positive side of any surface within the solid, an expression of the form

 ${\displaystyle \alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'}}$ (379)
represents the component parallel to ${\displaystyle X}$ of the force exerted upon the surface in the strained state by the matter on the positive side per unit of area measured in the state of reference. This is evident from the consideration that in estimating the force upon any surface we may substitute for the given surface a broken one consisting of elements for each of which either ${\displaystyle x'}$ or ${\displaystyle y'}$ or ${\displaystyle z'}$ is constant. Applied to a surface bounding a solid, or any portion of a solid which may not be continuous with the rest, when the normal is drawn outward as usual, the same expression taken negatively represents the component parallel to ${\displaystyle X}$ of the force exerted upon the surface (per unit of surface measured in the state of reference) by the interior of the solid, or of the portion considered. Equations (378) therefore express the condition that the force exerted upon the surface of discontinuity by the matter on one side and determined by its state of strain shall be equal and opposite to that exerted by the matter on the other side. Since
 ${\displaystyle (\alpha ')_{1}=-(\alpha ')_{2},}$⁠${\displaystyle (\beta ')_{1}=-(\beta ')_{2},}$⁠${\displaystyle (\gamma ')_{1}=-(\gamma ')_{2},}$
we may also write
 ${\displaystyle \alpha '(X_{X'})_{1}+\beta '(X_{Y'})_{1}+\gamma '(X_{Z'})_{1}=\alpha '(X_{X'})_{2}+\beta '(X_{Y'})_{2}+\gamma '(X_{Z'})_{2}}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}$ (380) etc.,

where the signs of ${\displaystyle \alpha ',\beta ',\gamma '}$ may be determined by the normal on either side of the surface of discontinuity.

Equation (371) expresses the mechanical condition of equilibrium for a surface where the solid meets a fluid. It involves the separate equations

 ${\displaystyle \alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'}=-\alpha p{\frac {Ds}{Ds'}},}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (381) ${\displaystyle \alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'}=-\beta p{\frac {Ds}{Ds'}},}$ ${\displaystyle \alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'}=-\gamma p{\frac {Ds}{Ds'}},}$

the fraction ${\displaystyle {\frac {Ds}{Ds'}}}$ denoting the ratio of the areas of the same element of the surface in the strained and unstrained states of the solid. These equations evidently express that the force exerted by the interior of the solid upon an element of its surface, and determined by the strain of the solid, must be normal to the surface and equal (but acting in the opposite direction) to the pressure exerted by the fluid upon the same element of surface.

If we wish to replace ${\displaystyle \alpha }$ and ${\displaystyle Ds}$ by ${\displaystyle \alpha ',\beta ',\gamma ',}$ and the quantities which express the strain of the element, we may make use of the following considerations. The product ${\displaystyle \alpha Ds}$ is the projection of the element ${\displaystyle Ds}$ on the Y-Z plane. Now since the ratio ${\displaystyle {\frac {Ds}{Ds'}}}$ is independent of the form of the element, we may suppose that it has any convenient form. Let it be bounded by the three surfaces ${\displaystyle x'={\text{const.}}}$, ${\displaystyle y'={\text{const.}}}$, ${\displaystyle z'={\text{const.}}}$, and let the parts of each of these surfaces included by the two others with the surface of the body be denoted by ${\displaystyle L,M,}$, and ${\displaystyle N}$, or by ${\displaystyle L',M',}$, and ${\displaystyle N'}$, according as we have reference to the strained or unstrained state of the body. The areas of ${\displaystyle L',M',}$, and ${\displaystyle N'}$ are evidently ${\displaystyle \alpha 'Ds',\beta 'Ds',}$, and ${\displaystyle \gamma 'Ds'}$; and the sum of the projections of ${\displaystyle L,M,}$, and ${\displaystyle N}$ upon any plane is equal to the projection of ${\displaystyle Ds}$ upon that plane, since ${\displaystyle L,M,}$, and ${\displaystyle N}$ with ${\displaystyle Ds}$ include a solid figure. (In propositions of this kind the sides of surfaces must be distinguished. If the normal to ${\displaystyle Ds}$ falls outward from the small solid figure, the normals to ${\displaystyle L,M,}$, and ${\displaystyle N}$ must fall inward, and vice versa.) Now ${\displaystyle L'}$ is a right-angled triangle of which the perpendicular sides may be called ${\displaystyle dy'}$ and ${\displaystyle dz'}$. The projection of ${\displaystyle L}$ on the Y-Z plane will be a triangle, the angular points of which are determined by the co-ordinates

 ${\displaystyle y,z;}$ ${\displaystyle y+{\frac {dy}{dy'}}dy',}$ ${\displaystyle z+{\frac {dz}{dy'}}dy';}$ ${\displaystyle y+{\frac {dy}{dz'}}dz',}$ ${\displaystyle z+{\frac {dz}{dz'}}dz';}$
the area of such a triangle is
 ${\displaystyle {\tfrac {1}{2}}\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)dy'dz',}$
or, since ${\displaystyle {\tfrac {1}{2}}dy'dz'}$ represents the area of ${\displaystyle L'}$,
 ${\displaystyle \left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)\alpha 'Ds'.}$
(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 ${\displaystyle M}$ and ${\displaystyle N}$ upon the same plane will be obtained by changing ${\displaystyle y',z'}$ and ${\displaystyle \alpha '}$ in this expression into ${\displaystyle z',x'}$, and ${\displaystyle \beta '}$, and into ${\displaystyle x',y',}$ and ${\displaystyle \gamma '}$. The sum of the three expressions may be substituted for a ${\displaystyle Ds}$ in (381).

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

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

 ${\displaystyle \textstyle \sum '\displaystyle (\alpha 'X_{X'})+p\textstyle \sum '\displaystyle \left\lbrace \alpha '\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)\right\rbrace =0,}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}$ (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, ${\displaystyle \delta N'}$ may have either positive or negative values, and we must have

 ${\displaystyle \epsilon _{V'}-t\eta _{V'}-pv_{V'}=\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}').}$ (383)
But if some of the components of the solid are only possible components (see page 64) of the fluid, ${\displaystyle \delta N'}$ is incapable of positive values, as the quantity of the solid cannot be increased, and it is sufficient for equilibrium that
 ${\displaystyle \epsilon _{V'}-t\eta _{V'}-pv_{V'}\leqq \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}').}$ (384)
To express- condition (383) in a form independent of the state of reference, we may use ${\displaystyle \epsilon _{V},\eta _{V},\Gamma _{1}}$, etc., to denote the densities of energy, of entropy, and of the several component substances in the variable state of the solid. We shall obtain, on dividing the equation by ${\displaystyle v_{V'}}$,
 ${\displaystyle \epsilon _{V}+t\eta _{V}+p=\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}).}$ (385)
It will be remembered that the summation relates to the several components of the solid. If the solid is of uniform composition throughout, or if we only care to consider the contact of the solid and the fluid at a single point, we may treat the solid as composed of a single substance. If we use ${\displaystyle \mu _{1}}$ to denote the potential for this substance in the fluid, and ${\displaystyle \Gamma }$ to denote the density of the solid in the variable state (${\displaystyle \Gamma '}$, as before denoting its density in the state of reference), we shall have
 ${\displaystyle \epsilon _{V'}+t\eta _{V'}+pv_{V'}=\mu _{1}\Gamma ',}$ (386)
 and⁠${\displaystyle \epsilon _{V}+t\eta _{V}+p=\mu _{1}\Gamma .}$ (387)
To fix our ideas in discussing this condition, let us apply it to the case of a solid body which is homogeneous in nature and in state of strain. If we denote by ${\displaystyle \epsilon ,\eta ,v,}$ and ${\displaystyle m}$, its energy, entropy, volume, and mass, we have
 ${\displaystyle \epsilon -t\eta +pv=\mu _{1}m.}$ (388)
Now the mechanical conditions of equilibrium for the surface where a solid meets a fluid require that the traction upon the surface determined by the state of strain of the solid shall be normal to the surface. This condition is always satisfied with respect to three surfaces at right angles to one another. In proving this well-known proposition, we shall lose nothing in generality, if we make the state of reference, which is arbitrary, coincident with the state under discussion, the axes to which these states are referred being also coincident. We shall then have, for the normal component of the traction per unit of surface across any surface for which the direction-cosines of the normal are ${\displaystyle \alpha ,\beta ,\gamma }$ (compare (379), and for the notation ${\displaystyle X_{X}}$, etc., page 190),
 ${\displaystyle S=}$ ${\displaystyle \alpha (\alpha X_{X}+\beta X_{Y}+\gamma X_{Z})}$ ${\displaystyle +\beta (\alpha Y_{X}+\beta Y_{Y}+\gamma Y_{Z})}$ ${\displaystyle +\gamma (\alpha Z_{X}+\beta Z_{Y}+\gamma Z_{Z}),}$

or, by (375), (376),

 ${\displaystyle S=\alpha ^{2}X_{X}+\beta ^{2}Y_{X}+\gamma ^{2}Z_{X}+2\alpha \beta X_{Y}+2\beta \gamma Y_{Z}+2\gamma \alpha Z_{X}.}$ (389)
We may also choose any convenient directions for the co-ordinate axes. Let us suppose that the direction of the axis of ${\displaystyle X}$ is so chosen that the value of ${\displaystyle S}$ for the surface perpendicular to this axis is as great as for any other surface, and that the direction of the axis of ${\displaystyle Y}$ (supposed at right angles to ${\displaystyle X}$) is such that the value of ${\displaystyle S}$ for the surface perpendicular to it is as great as for any other surface passing through the axis of ${\displaystyle X}$. Then, if we write ${\displaystyle {\frac {dS}{d\alpha }}}$, ${\displaystyle {\frac {dS}{d\beta }}}$, ${\displaystyle {\frac {dS}{d\gamma }}}$ for the differential coefficients derived from the last equation by treating ${\displaystyle \alpha ,\beta }$ and ${\displaystyle \gamma }$ as independent variables,
 ${\displaystyle {\frac {Ds}{d\alpha }}d\alpha +{\frac {dS}{d\beta }}d\beta +{\frac {dS}{d\gamma }}d\gamma =0,}$

 when ${\displaystyle \alpha d\alpha +\beta d\beta +\gamma d\gamma =0,}$ and ${\displaystyle \alpha =1,}$⁠${\displaystyle \beta =0,}$⁠${\displaystyle \gamma =0,}$ That is, ${\displaystyle {\frac {dS}{d\beta }}=0,}$⁠and⁠${\displaystyle {\frac {dS}{d\gamma }}=0,}$ when ${\displaystyle \alpha =1,}$⁠${\displaystyle \beta =0,}$⁠${\displaystyle \gamma =0.}$ Hence ${\displaystyle X_{Y}=0,}$⁠and⁠${\displaystyle Z_{X}=0.}$ (390) Moreover, ${\displaystyle {\frac {dS}{d\beta }}d\beta +{\frac {dS}{d\gamma }}d\gamma =0,}$ when ${\displaystyle \alpha =0,}$⁠${\displaystyle d\alpha =0,}$ ${\displaystyle \beta d\beta +\gamma d\gamma =0,}$ and ${\displaystyle \beta =1,}$⁠${\displaystyle \gamma =0.}$ Hence ${\displaystyle Y_{Z}=0.}$ (391)

Therefore, when the co-ordinate axes have the supposed directions, which are called the principal axes of stress, the rectangular components of the traction across any surface (${\displaystyle \alpha ,\beta ,\gamma }$) are by (379)

 ${\displaystyle \alpha X_{X},}$ ${\displaystyle \beta Y_{T},}$ ${\displaystyle \gamma Z_{Z}.}$ (392)
Hence, the traction across any surface will be normal to that surface,—

(1), when the surface is perpendicular to a principal axis of stress;

(2), if two of the principal tractions ${\displaystyle X_{X},Y_{Y},Z_{Z}}$ are equal, when the surface is perpendicular to the plane containing the two corresponding axes (in this case the traction across any such surface is equal to the common value of the two principal tractions);

(3), if the principal tractions are all equal, the traction is normal and constant for all surfaces.

It will be observed that in the second and third cases the positions of the principal axes of stress are partially or wholly indeterminate (so that these cases may be regarded as included in the first), but the values of the principal tractions are always determinate, although not always different.

If, therefore, a solid which is homogeneous in nature and in state of strain is bounded by six surfaces perpendicular to the principal axes of stress, the mechanical conditions of equilibrium for these surfaces may be satisfied by the contact of fluids having the proper pressures (see (381)), which will in general be different for the different pairs of opposite sides, and may be denoted by ${\displaystyle p',p'',p'''}$. (These pressures are equal to the principal tractions of the solid taken negatively.) It will then be necessary for equilibrium with respect to the tendency of the solid to dissolve that the potential for the substance of the solid in the fluids shall have values ${\displaystyle \mu _{1}',\mu _{1}'',\mu _{1}''',}$ determined by the equations

 ${\displaystyle \epsilon -t\eta +p'v=\mu _{1}'m,}$ (393)
 ${\displaystyle \epsilon -t\eta +p''v=\mu _{1}''m,}$ (394)
 ${\displaystyle \epsilon -t\eta +p'''v=\mu _{1}'''m.}$ (395)
These values, it will be observed, are entirely determined by the nature and state of the solid, and their differences are equal to the differences of the corresponding pressures divided by the density of the solid.

It may be interesting to compare one of these potentials, as ${\displaystyle \mu _{1}'}$, with the potential (for the same substance) in a fluid of the same temperature ${\displaystyle t}$ and pressure ${\displaystyle p'}$ which would be in equilibrium with the same solid subjected on all sides to the uniform pressure ${\displaystyle p'}$. If we write ${\displaystyle [\epsilon ]_{p'},[\eta ]_{p'},[v]_{p'},}$ and ${\displaystyle [\mu _{1}]_{p'}}$ for the values which ${\displaystyle \epsilon ,\eta ,v,}$ and ${\displaystyle \mu _{1}}$ would receive on this supposition, we shall have

 ${\displaystyle [\epsilon ]_{p'}-t[\eta ]_{p'}+p'[v]_{p'}=[\mu _{1}]_{p'}m.}$ (396)
Subtracting this from (393), we obtain
 ${\displaystyle \epsilon -[\epsilon ]_{p'}-t\eta +t[\eta ]_{p'}+p'v-p'[v]_{p'}=\mu _{1}m-[\mu _{1}]_{p'}m.}$ (397)
Now it follows immediately from the definitions of energy and entropy that the first four terms of this equation represent the work spent upon the solid in bringing it from the state of hydrostatic stress to the other state without change of temperature, and ${\displaystyle p'v-p'[v]_{p'}}$ evidently denotes the work done in displacing a fluid of pressure ${\displaystyle p'}$ surrounding the solid during the operation. Therefore, the first number of the equation represents the total work done in bringing the solid when surrounded by a fluid of pressure ${\displaystyle p'}$ from the state of hydrostatic stress ${\displaystyle p'}$ to the state of stress ${\displaystyle p',p'',p'''}$. This quantity is necessarily positive, except of course in the limiting case when ${\displaystyle p'=p''=p'''}$. If the quantity of matter of the solid body be unity, the increase of the potential in the fluid on the side of the solid on which the pressure remains constant, which will be necessary to maintain equilibrium, is equal to the work done as above described. Hence, ${\displaystyle \mu _{1}'}$ is greater than ${\displaystyle [\mu _{1}]_{p'}}$, and for similar reasons ${\displaystyle \mu _{1}''}$ is greater than the value of the potential which would be necessary for equilibrium if the solid were subjected to the uniform pressure ${\displaystyle p''}$, and ${\displaystyle \mu _{1}'''}$ greater than that which would be necessary for equilibrium if the solid were subjected to the uniform pressure ${\displaystyle p'''}$. That is (if we adapt our language to what we may regard as the most general case, viz., that in which the fluids contain the substance of the solid but are not wholly composed of that substance), the fluids in equilibrium with the solid are all supersaturated with respect to the substance of the solid, except when the solid is in a state of hydrostatic stress; so that if there were present in any one of these fluids any small fragment of the same kind of solid subject to the hydrostatic pressure of the fluid, such a fragment would tend to increase. Even when no such fragment is present, although there must be perfect equilibrium so far as concerns the tendency of the solid to dissolve or to increase by the accretion of similarly strained matter, yet the presence of the solid which is subject to the distorting stresses, will doubtless facilitate the commencement of the formation of a solid of hydrostatic stress upon its surface, to the same extent, perhaps, in the case of an amorphous body, as if it were itself subject only to hydrostatic stress. This may sometimes, or perhaps generally, make it a necessary condition of equilibrium in cases of contact between a fluid and an amorphous solid which can be formed out of it, that the solid at the surface where it meets the fluid shall be sensibly in a state of hydrostatic stress.

But in the case of a solid of continuous crystalline structure, subjected to distorting stresses and in contact with solutions satisfying the conditions deduced above, although crystals of hydrostatic stress would doubtless commence to form upon its surface (if the distorting stresses and consequent supersaturation of the fluid should be carried too far), before they would commence to be formed within the fluid or on the surface of most other bodies, yet within certain limits the relations expressed by equations (393)–(395) must admit of realization, especially when the solutions are such as can be easily supersaturated.[2]

It may be interesting to compare the variations of ${\displaystyle p}$, the pressure in the fluid which determines in part the stresses and the state of strain of the solid, with other variations of the stresses or strains in the solid, with respect to the relation expressed by equation (388). To examine this point with complete generality, we may proceed in the following manner.

Let us consider so much of the solid as has in the state of reference the form of a cube, the edges of which are equal to unity, and parallel to the co-ordinate axes. We may suppose this body to be homogeneous in nature and in state of strain both in its state of reference and in its variable state. (This involves no loss of generality, since we may make the unit of length as small as we choose.) Let the fluid meet the solid on one or both of the surfaces for which ${\displaystyle Z'}$ is constant. We may suppose these surfaces to remain perpendicular to the axis of ${\displaystyle Z}$ in the variable state of the solid, and the edges in which ${\displaystyle y'}$ and ${\displaystyle z'}$ are both constant to remain parallel to the axis of ${\displaystyle X}$. It will be observed that these suppositions only fix the position of the strained body relatively to the co-ordinate axes, and do not in any way limit its state of strain.

It follows from the suppositions which we have made that

 ${\displaystyle {\frac {dz}{dx'}}={\text{const.}}=0,}$⁠${\displaystyle {\frac {dz}{dy'}}={\text{const.}}=0,}$⁠${\displaystyle {\frac {dy}{dx'}}={\text{const.}}=0;}$ (398)
 and⁠${\displaystyle X_{Z'}=0,}$⁠${\displaystyle Y_{Z'}=0,}$⁠${\displaystyle Z_{Z'}=-p{\frac {dx}{dx'}}{\frac {dy}{dy'}}\cdot }$ (399)
Hence, by (355),
 ${\displaystyle d\epsilon _{V'}=td\eta _{V'}+X_{X'}d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+Y_{Y'}d{\frac {dy}{dy'}}-p{\frac {dx}{dx'}}{\frac {dy}{dy'}}d{\frac {dz}{dz'}}\cdot }$ (400)
Again, by (388),
 ${\displaystyle d\epsilon =td\eta +\eta dt-pdv-vdp+md\mu _{1}.}$ (401)
Now the suppositions which have been made require that
 ${\displaystyle v={\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}},}$ (402)
 and⁠${\displaystyle dv={\frac {dy}{dy'}}{\frac {dz}{dz'}}d{\frac {dx}{dx'}}+{\frac {dz}{dz'}}{\frac {dx}{dx'}}d{\frac {dy}{dy'}}+{\frac {dx}{dx'}}{\frac {dy}{dy'}}d{\frac {dz}{dz'}}\cdot }$ (403)
Combining equations (400), (401), and (403), and observing that ${\displaystyle \epsilon _{V'}}$ and ${\displaystyle \eta _{V'}}$ are equivalent to ${\displaystyle \epsilon }$ and ${\displaystyle \eta }$, we obtain
 ${\displaystyle \eta dt-vdp+md\mu _{1}=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot }$ (404)
The reader will observe that when the solid is subjected on all sides to the uniform normal pressure ${\displaystyle p}$, the coefficients of the differentials in the second member of this equation will vanish. For the expression ${\displaystyle {\frac {dy}{dy'}}{\frac {dz}{dz'}}}$ represents the projection on the Y-Z plane of a side of the parallelepiped for which ${\displaystyle x'}$ is constant, and multiplied by ${\displaystyle p}$ it will be equal to the component parallel to the axis of ${\displaystyle X}$ of the total pressure across this side, i.e., it will be equal to ${\displaystyle X_{X'}}$ taken negatively. The case is similar with respect to the coefficient of ${\displaystyle d{\frac {dy}{dy'}};}$ and ${\displaystyle X_{Y'}}$ evidently denotes a force tangential to the surface on which it acts. It will also be observed, that if we regard the forces acting upon the sides of the solid parallelepiped as composed of the hydrostatic pressure ${\displaystyle p}$ together with additional forces, the work done in any infinitesimal variation of the state of strain of the solid by these additional forces will be represented by the second member of the equation.

We will first consider the case in which the fluid is identical in substance with the solid. We have then, by equation (97), for a mass of the fluid equal to that of the solid,

 ${\displaystyle \eta _{F}dt-v_{F}dp+md\mu _{1}=0,}$ (405)
${\displaystyle \eta _{F}}$ and ${\displaystyle v_{F}}$ denoting the entropy and volume of the fluid. By subtraction we obtain
 ${\displaystyle -(\eta _{F}-\eta )dt+(v_{F}-v)dp=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot }$ (406)
Now if the quantities ${\displaystyle {\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}}$ remain constant, we shall have for the relation between the variations of temperature and pressure which is necessary for the preservation of equilibrium
 ${\displaystyle {\frac {dy}{dp}}={\frac {v_{F}-v}{\eta _{F}-\eta }}=t{\frac {v_{F}-v}{Q}},}$ (407)
where ${\displaystyle Q}$ denotes the heat which would be absorbed if the solid body should pass into the fluid state without change of temperature or pressure. This equation is similar to (131), which applies to bodies subject to hydrostatic pressure. But the value of ${\displaystyle {\frac {dt}{dp}}}$ will not generally be the same as if the solid were subject on all sides to the uniform normal pressure ${\displaystyle p}$; for the quantities ${\displaystyle v}$ and ${\displaystyle \eta }$ (and therefore ${\displaystyle Q}$) will in general have different values. But when the pressures on all sides are normal and equal, the value of ${\displaystyle {\frac {dt}{dp}}}$ will be the same, whether we consider the pressure when varied as still normal and equal on all sides, or consider the quantities ${\displaystyle {\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}}$ as constant.

But if we wish to know how the temperature is affected if the pressure between the solid and fluid remains constant, but the strain of the solid is varied in any way consistent with this supposition, the differential coefficients of ${\displaystyle t}$ with respect to the quantities which express the strain are indicated by equation (406). These differential coefficients all vanish, when the pressures on all sides are normal and equal, but the differential coefficient ${\displaystyle {\frac {dt}{dp}}}$, when ${\displaystyle {\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}}$ are constant, or when the pressures on all sides are normal and equal, vanishes only when the density of the fluid is equal to that of the solid.

The case is nearly the same when the fluid is not identical in substance with the solid, if we suppose the composition of the fluid to remain unchanged. We have necessarily with respect to the fluid

 ${\displaystyle d\mu _{1}=\left({\frac {d\mu _{1}}{dt}}\right)_{p,m}^{(F)}dt+\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}dp,}$[3] (408)
where the index (f) is used to indicate that the expression to which it is affixed relates to the fluid. But by equation (92)
 ${\displaystyle \left({\frac {d\mu _{1}}{dt}}\right)_{p,m}^{(F)}=-\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}}$,⁠and⁠${\displaystyle \left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}\cdot }$ (409)
Substituting these values in the preceding equation, transposing terms, and multiplying by ${\displaystyle m}$, we obtain
 ${\displaystyle m\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}-m\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}dp+md\mu _{1}=0.}$ (410)
By subtracting this equation from (404) we may obtain an equation similar to (406), except that in place of ${\displaystyle \eta _{F}}$ and ${\displaystyle v_{F}}$ we shall have the expressions
 ${\displaystyle m\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}}$⁠and⁠${\displaystyle m\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}\cdot }$
The discussion of equation (406) will therefore apply mutatis mutandis to this case.

We may also wish to find the variations in the composition of the fluid which will be necessary for equilibrium when the pressure ${\displaystyle p}$ or the quantities ${\displaystyle {\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}}$ are varied, the temperature remaining constant. If we know the value for the fluid of the quantity represented by ${\displaystyle \xi }$ on page 87 in terms of ${\displaystyle t,p}$, and the quantities of the several components ${\displaystyle m_{1},m_{2},m_{3}}$, etc., the first of which relates to the substance of which the solid is formed, we can easily find the value of ${\displaystyle \mu _{1}}$ in terms of the same variables. Now in considering variations in the composition of the fluid, it will be sufficient if we make all but one of the components variable. We may therefore give to ${\displaystyle m_{1}}$ a constant value, and making ${\displaystyle t}$ also constant, we shall have

 ${\displaystyle d\mu _{1}=\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}dp+\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,p,m}^{(F)}dm_{2}+\left({\frac {d\mu _{1}}{dm_{3}}}\right)_{t,p,m}^{(F)}dm_{3}+{\text{etc.}}}$
Substituting this value in equation (404), and cancelling the term containing ${\displaystyle dt}$, we obtain
 ${\displaystyle \left\{m\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}-v\right\}\quad dp+m\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,p,m}^{(F)}dm_{2}+m\left({\frac {d\mu _{1}}{dm_{3}}}\right)_{t,p,m}^{(F)}dm_{3}+{\text{etc.}}=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dz'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot }$ (411)
This equation shows the variation in the quantity of any one of the components of the fluid (other than the substance which forms the solid) which will balance a variation of ${\displaystyle p,}$ or of ${\displaystyle {\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dz}{dz'}}}$ with respect to the tendency of the solid to dissolve.

Fundamental Equations for Solids.

The principles developed in the preceding pages show that the solution of problems relating to the equilibrium of a solid, or at least their reduction to purely analytical processes, may be made to depend upon our knowledge of the composition and density of the solid at every point in some particular state, which we have called the state of reference, and of the relation existing between the quantities which have been represented by ${\displaystyle \epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}},x',y',}$ and ${\displaystyle z'.}$ When the solid is in contact with fluids, a certain knowledge of the properties of the fluids is also requisite, but only such as is necessary for the solution of problems relating to the equilibrium of fluids among themselves.

If in any state of which a solid is capable, it is homogeneous in its nature and in its state of strain, we may choose this state as the state of reference, and the relation between ${\displaystyle \epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}}}$ will be independent of ${\displaystyle x',y',z'.}$ But it is not always possible, even in the case of bodies which are homogeneous in nature, to bring all the elements simultaneously into the same state of strain. It would not be possible, for example, in the case of a Prince Rupert's drop.

If, however, we know the relation between ${\displaystyle \epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}}}$ for any kind of homogeneous solid, with respect to any given state of reference, we may derive from it a similar relation with respect to any other state as a state of reference. For if ${\displaystyle x',y',z'}$ denote the co-ordinates of points of the solid in the first state of reference, and ${\displaystyle x'',y'',z''}$ the co-ordinates of the same points in the second state of reference, we shall have necessarily

 ${\displaystyle {\frac {dx}{dx'}}={\frac {dx}{dx''}}{\frac {dx''}{dx'}}+{\frac {dx}{dy''}}{\frac {dy''}{dx'}}+{\frac {dx}{dz''}}{\frac {dz''}{dx'}},}$ etc. (nine equations), (412)
and if we write ${\displaystyle R}$ for the volume of an element in the state ${\displaystyle (x'',y'',z'')}$ divided by its volume in the state ${\displaystyle (x',y',z')}$ we shall have
 ${\displaystyle R={\begin{vmatrix}{\frac {dx''}{dx'}}&{\frac {dx''}{dy'}}&{\frac {dx''}{dz'}}\\{\frac {dy''}{dx'}}&{\frac {dy''}{dy'}}&{\frac {dy''}{dz'}}\\{\frac {dz''}{dx'}}&{\frac {dz''}{dy'}}&{\frac {dz''}{dz'}}\end{vmatrix}},}$ (413)
 ${\displaystyle \epsilon _{V'}=R\epsilon _{V''},}$⁠${\displaystyle \eta _{V'}=R\eta _{V''}.}$ (414)
If, then, we have ascertained by experiment the value of ${\displaystyle \epsilon _{V'}}$ in terms of ${\displaystyle \eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}},}$ and the quantities which express the composition of the body, by the substitution of the values given in (412)–(414), we shall obtain ${\displaystyle \epsilon _{V''}}$ terms of ${\displaystyle \eta _{V''},{\frac {dx}{dx''}},...{\frac {dz}{dz''}},{\frac {dx''}{dx'}},...{\frac {dz''}{dz'}}}$ and the quantities which express the composition of the body.

We may apply this to the elements of a body which may be variable from point to point in composition and state of strain in a given state of reference ${\displaystyle (x'',y'',z'')}$, and if the body is fully described in that state of reference, both in respect to its composition and to the displacement which it would be necessary to give to a homogeneous solid of the same composition, for which ${\displaystyle \epsilon _{V'}}$ is known in terms of ${\displaystyle \eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$ and the quantities which express its composition, to bring it from the state of reference ${\displaystyle (x',y',z')}$ into a similar and similarly situated state of strain with that of the element of the non-homogeneous body, we may evidently regard ${\displaystyle {\frac {dx''}{dx'}},...{\frac {dz''}{dz'}}}$ as known for each element of the body, that is, as known in terms of ${\displaystyle x'',y'',z''.}$ We shall then have ${\displaystyle \epsilon _{V''}}$ in terms of ${\displaystyle \eta _{V''},{\frac {dx}{dx''}},...{\frac {dz}{dz''}},x'',y'',z''}$; and since the composition of the body is known in terms of ${\displaystyle x'',y'',z''}$, and the density, if not given directly, can be determined from the density of the homogeneous body in its state of reference ${\displaystyle (x',y',z')}$, this is sufficient for determining the equilibrium of any given state of the non-homogeneous solid.

An equation, therefore, which expresses for any kind of solid, and with reference to any determined state of reference, the relation between the quantities denoted by ${\displaystyle \epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$, involving also the quantities which express the composition of the body, when that is capable of continuous variation, or any other equation from which the same relations may be deduced, may be called a fundamental equation for that kind of solid. It will be observed that the sense in which this term is here used, is entirely analogous to that in which we have already applied the term to fluids and solids which are subject only to hydrostatic pressure.

When the fundamental equation between ${\displaystyle \epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$ is known, we may obtain by differentiation the values of ${\displaystyle t,X_{X'},...Z_{Z'}}$ in terms of the former quantities, which will give eleven independent relations between the twenty-one quantities

 ${\displaystyle \epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}},t,X_{X'},...Z_{Z'},}$ (415)
which are all that exist, since ten of these quantities are independent. All these equations may also involve variables which express the composition of the body, when that is capable of continuous variation.

If we use the symbol ${\displaystyle \psi _{V'}}$ to denote the value of ${\displaystyle \psi }$ (as defined on page 89) for any element of a solid divided by the volume of the element in the state of reference, we shall have

 ${\displaystyle \psi _{V'}=\epsilon _{V'}-t\eta _{V'}.}$ (416)
The equation (356) may be reduced to the form
 ${\displaystyle \delta \psi _{V'}=-\eta _{V'}\delta t+\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)\cdot }$ (417)
Therefore, if we know the value of ${\displaystyle \psi _{V'}}$ in terms of the variables ${\displaystyle t,{\frac {dx}{dx'}},...{\frac {dz}{dz'}},}$ together with those which express the composition of the body, we may obtain by differentiation the values of ${\displaystyle \eta _{V'},X_{X'},...Z_{Z'},}$ in terms of the same variables. This will make eleven independent relations between the same quantities as before, except that we shall have ${\displaystyle \psi _{V'}}$ instead of ${\displaystyle \epsilon _{V'}}$. Or if we eliminate ${\displaystyle \psi _{V'}}$ by means of equation (416), we shall obtain eleven independent equations between the quantities in (415) and those which express the composition of the body. An equation, therefore, which determines the value of ${\displaystyle \psi _{V'}}$ as a function of the quantities ${\displaystyle t,{\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$ and the quantities which express the composition of the body when it is capable of continuous variation, is a fundamental equation for the kind of solid to which it relates.

In the discussion of the conditions of equilibrium of a solid, we might have started with the principle that it is necessary and sufficient for equilibrium that the temperature shall be uniform throughout the whole mass in question, and that the variation of the force-function (${\displaystyle -\psi }$) of the same mass shall be null or negative for any variation in the state of the mass not affecting its temperature. We might have assumed that the value of ${\displaystyle \psi }$ for any same element of the solid is a function of the temperature and the state of strain, so that for constant temperature we might write

 ${\displaystyle \delta \psi _{V'}=\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right),}$
the quantities ${\displaystyle X_{X'},...Z_{Z'}}$, being defined by this equation. This would be only a formal change in the definition of ${\displaystyle X_{X'},...Z_{Z'}}$ and would not affect their values, for this equation holds true of ${\displaystyle X_{X'},...Z_{Z'}}$ as defined by equation (355). With such data, by transformations similar to those which we have employed, we might obtain similar results.[4] It is evident that the only difference in the equations would be that ${\displaystyle \psi _{V'}}$ would take the place of ${\displaystyle \epsilon _{V'}}$, and that the terms relating to entropy would be wanting. Such a method is evidently preferable with respect to the directness with which the results are obtained. The method of this paper shows more distinctly the rôle of energy and entropy in the theory of equilibrium, and can be extended more naturally to those dynamical problems in which motions take place under the condition of constancy of entropy of the elements of a solid (as when vibrations are propagated through a solid), just as the other method can be more naturally extended to dynamical problems in which the temperature is constant. (See ,note on page 90.) We have already had occasion to remark that the state of strain of any element considered without reference to directions in space is capable of only six independent variations. Hence, it must be possible to express the state of strain of an element by six functions of ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$, which are independent of the position of the element. For these quantities we may choose the squares of the ratios of elongation of lines parallel to the three co-ordinate axes in the state of reference, and the products of the ratios of elongation for each pair of these lines multiplied by the cosine of the angle which they include in the variable state of the solid. If we denote these quantities by ${\displaystyle A,B,C,a,b,c}$ we shall have
 ${\displaystyle A=\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2},}$ ${\displaystyle B=\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2},}$ ${\displaystyle C=\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2},}$ (418) ${\displaystyle a=\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dz'}}\right),}$ ${\displaystyle b=\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right),}$ ${\displaystyle c=\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\cdot }$ (419)

The determination of the fundamental equation for a solid is thus reduced to the determination of the relation between ${\displaystyle \epsilon _{V'},\eta _{V'},A,B,C,a,b,c,}$ or of the relation between ${\displaystyle \psi _{V'},t,A,B,C,a,b,c}$.

In the case of isotropic solids, the state of strain of an element, so far as it can affect the relation of ${\displaystyle \epsilon _{V'}}$ and ${\displaystyle \eta _{V'}}$ or of ${\displaystyle \psi _{V'}}$ and ${\displaystyle t}$, is capable of only three independent variations. This appears most distinctly as a consequence of the proposition that for any given strain of an element there are three lines in the element which are at right angles to one another both in its unstrained and in its strained state. If the unstrained element is isotropic, the ratios of elongation for these three lines must with ${\displaystyle \eta _{V'}}$ determine the value ${\displaystyle \epsilon _{V'}}$, or with ${\displaystyle t}$ determine the value of ${\displaystyle \psi _{V'}}$.

To demonstrate the existence of such lines, which are called the principal axes of strain, and to find the relations of the elongations of such lines to the quantities ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$, we may proceed as follows. The ratio of elongation ${\displaystyle r}$ of any line of which ${\displaystyle \alpha ',\beta ',\gamma '}$ are the direction-cosines in the state of reference is evidently given by the equation

 ${\displaystyle r^{2}}$ ${\displaystyle =\left({\frac {dx}{dx'}}\alpha '+{\frac {dx}{dy'}}\beta '+{\frac {dx}{dz'}}\gamma '\right)^{2},}$ ${\displaystyle +\left({\frac {dy}{dx'}}\alpha '+{\frac {dy}{dy'}}\beta '+{\frac {dy}{dz'}}\gamma '\right)^{2},}$ ${\displaystyle +\left({\frac {dz}{dx'}}\alpha '+{\frac {dz}{dy'}}\beta '+{\frac {dz}{dz'}}\gamma '\right)^{2}\cdot }$

Now the proposition to be established is evidently equivalent to this—that it is always possible to give such directions to the two systems of rectangular axes ${\displaystyle X',Y',Z'}$, and ${\displaystyle X,Y,Z}$, that

 ${\displaystyle {\frac {dx}{dy'}}=0,}$ ${\displaystyle {\frac {dx}{dz'}}=0,}$ ${\displaystyle {\frac {dy}{dz'}}=0,}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}}$ (421) ${\displaystyle {\frac {dy}{dx'}}=0,}$ ${\displaystyle {\frac {dz}{dx'}}=0,}$ ${\displaystyle {\frac {dz}{dy'}}=0.}$

We may choose a line in the element for which the value of ${\displaystyle r}$ is at least as great as for any other, and make the axes of ${\displaystyle X}$ and ${\displaystyle X'}$ parallel to this line in the strained and unstrained states respectively.

Then

${\displaystyle {\frac {dy}{dx'}}=0,}$${\displaystyle {\frac {dz}{dx'}}=0.}$

(422)

Moreover, if we write ${\displaystyle {\frac {d(r^{2})}{d\alpha '}},{\frac {d(r^{2})}{d\beta '}},{\frac {d(r^{2})}{d\gamma '}}}$ for the differential coefficients obtained from (420) by treating ${\displaystyle \alpha ',\beta ',\gamma '}$ as independent variables,
 ${\displaystyle {\frac {d(r^{2})}{d\alpha '}}+{\frac {d(r^{2})}{d\beta '}}+{\frac {d(r^{2})}{d\gamma '}}=0,}$ when ${\displaystyle \alpha 'd\alpha '+\beta 'd\beta '+\gamma 'd\gamma '=0,}$ and ${\displaystyle \alpha '=1,}$⁠${\displaystyle \beta '=0,}$⁠${\displaystyle \gamma '=0.}$ That is, ${\displaystyle {\frac {d(r^{2})}{d\beta '}}=0,}$⁠and⁠${\displaystyle {\frac {d(r^{2})}{d\gamma '}}=0,}$ when ${\displaystyle \alpha '=1,}$⁠${\displaystyle \beta '=0,}$⁠${\displaystyle \gamma '=0.}$ Hence, ${\displaystyle {\frac {dx}{dy'}}=0,}$⁠${\displaystyle {\frac {dx}{dz'}}=0.}$

Therefore a line of the element which in the unstrained state is perpendicular to ${\displaystyle X'}$ is perpendicular to ${\displaystyle X}$ in the strained state. Of all such lines we may choose one for which the value of ${\displaystyle r}$ is at least as great as for any other, and make the axes of ${\displaystyle Y'}$ and ${\displaystyle Y}$ parallel to this line in the unstrained and in the strained state respectively. Then

 ${\displaystyle {\frac {dz}{dy'}}=0;}$ (424)
and it may easily be shown by reasoning similar to that which has just been employed that
 ${\displaystyle {\frac {dy}{dz'}}=0.}$ (425)
Lines parallel to the axes of ${\displaystyle X',Y'}$, and ${\displaystyle Z'}$ in the unstrained body will therefore be parallel to ${\displaystyle X,Y}$, and ${\displaystyle Z}$ in the strained body, and the ratios of elongation for such lines will be
 ${\displaystyle {\frac {dx}{dx'}},{\frac {dy}{dy'}},{\frac {dz}{dz'}}\cdot }$
These lines have the common property of a stationary value of the ratio of elongation for varying directions of the line. This appears from the form to which the general value of ${\displaystyle r^{2}}$ is reduced by the positions of the co-ordinate axes, viz.,
 ${\displaystyle r^{2}={\frac {dx}{dx'}}\alpha ^{'2}+{\frac {dy}{dy'}}\beta ^{'2}+{\frac {dz}{dz'}}\gamma ^{'2}.}$
Having thus proved the existence of lines, with reference to any particular strain, which have the properties mentioned, let us proceed to find the relations between the ratios of elongation for these lines (the principal axes of strain) and the quantities ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$ under the most general supposition with respect to the position of the co-ordinate axes.

For any principal axis of strain we have

 ${\displaystyle {\frac {d(r^{2})}{d\alpha '}}d\alpha '+{\frac {d(r^{2})}{d\beta '}}d\beta '+{\frac {d(r^{2})}{d\gamma '}}d\gamma '=0,}$
 when⁠${\displaystyle \alpha 'd\alpha '+\beta '\beta '+\gamma 'd\gamma '=0,}$
the differential coefficients in the first of these equations being determined from (420) as before. Therefore,
 ${\displaystyle {\frac {1}{\alpha '}}{\frac {d(r^{2})}{d\alpha '}}={\frac {1}{\beta '}}{\frac {d(r^{2})}{d\beta '}}={\frac {1}{\gamma '}}{\frac {d(r^{2})}{d\gamma '}}\cdot }$ (426)
From (420) we obtain directly
 ${\displaystyle {\frac {\alpha '}{2}}{\frac {d(r^{2})}{d\alpha '}}+{\frac {\beta '}{2}}{\frac {d(r^{2})}{d\beta '}}+{\frac {\gamma '}{2}}{\frac {d(r^{2})}{d\gamma '}}=r^{2}.}$ (427)
From the two last equations, in virtue of the necessary relation ${\displaystyle \alpha ^{'2}+\beta ^{'2}+\gamma ^{'2}=1}$, we obtain
 ${\displaystyle {\tfrac {1}{2}}{\frac {d(r^{2})}{d\alpha '}}=\alpha 'r^{2},}$⁠${\displaystyle {\tfrac {1}{2}}{\frac {d(r^{2})}{d\beta '}}=\beta 'r^{2},}$⁠${\displaystyle {\tfrac {1}{2}}{\frac {d(r^{2})}{d\gamma '}}=\gamma 'r^{2},}$ (428)
or, if we substitute the values of the differential coefficients taken from (420),
 ${\displaystyle \alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)=\alpha 'r^{2},}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (429) ${\displaystyle \alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dx'}}\right)+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)^{2}=\beta 'r^{2},}$ ${\displaystyle \alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right)+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dy'}}\right)+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2}=\gamma 'r^{2}.}$

If we eliminate ${\displaystyle \alpha ',\beta ',\gamma '}$ from these equations, we may write the result in the form,

 ${\displaystyle {\begin{vmatrix}\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}-r^{2}&\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)\\\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dx'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-r^{2}&\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dz'}}\right)\\\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dy'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2}-r^{2}\end{vmatrix}}=0.}$ (430)
We may write
 ${\displaystyle -r^{5}+Er^{4}-Fr^{2}+G=0.}$ (431)
 ${\displaystyle E=\textstyle \sum '\sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}\cdot }$ (432)
Also[5]
 ${\displaystyle F}$ ${\displaystyle =\textstyle \sum '\displaystyle \left\{\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\right\}\quad }$ ${\displaystyle =\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\right\}\quad }$ ${\displaystyle =\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\left({\frac {dy}{dy'}}\right)^{2}+\left({\frac {dy}{dx'}}\right)^{2}\left({\frac {dz}{dy'}}\right)^{2}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dy}{dx'}}{\frac {dy}{dy'}}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dz}{dx'}}{\frac {dz}{dy'}}\right\}\quad }$ ${\displaystyle =\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\left({\frac {dy}{dy'}}\right)^{2}+\left({\frac {dy}{dx'}}\right)^{2}\left({\frac {dx}{dy'}}\right)^{2}-2{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dy}{dx'}}{\frac {dy}{dy'}}\right\}\quad }$ ${\displaystyle =\textstyle \sum '\sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dy}{dx'}}{\frac {dx}{dy'}}\right)^{2}\cdot }$ (433)

This may also be written

 ${\displaystyle F=\textstyle \sum '\sum \displaystyle {\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}\end{vmatrix}}^{2}\cdot }$ (434)
In the reduction of the value of ${\displaystyle G}$, it will be convenient to use the symbol ${\displaystyle \textstyle \sum _{3+3}}$ to denote the sum of the six terms formed by changing ${\displaystyle x,y,z,}$ into ${\displaystyle y,z,x;}$${\displaystyle z,x,y;}$${\displaystyle x,z,y;}$${\displaystyle y,x,z;}$ and ${\displaystyle z,y,x}$ and the symbol ${\displaystyle \textstyle \sum _{3-3}}$ in the same sense except that the last three terms are to be taken negatively; also to use ${\displaystyle \textstyle \sum _{3-3}'}$ in a similar sense with respect to ${\displaystyle x',y',z'}$; and to use ${\displaystyle {\text{x', y', z'}}}$ as equivalent to ${\displaystyle x',y',z'}$ except that they are not to be affected by the sign of summation. With this understanding we may write
 ${\displaystyle G=\textstyle \sum _{3-3}'\left\{\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{x}}'}}{\frac {dx}{dx'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{y}}'}}{\frac {dx}{dy'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{z}}'}}{\frac {dx}{dz'}}\right)\right\}\quad \cdot }$ (435)
In expanding the product of the three sums, we may cancel on account of the sign ${\displaystyle \textstyle \sum _{3-3}'}$ the terms which do not contain all the three expressions ${\displaystyle dx,dy,}$ and ${\displaystyle dz}$. Hence we may write
 ${\displaystyle G}$ ${\displaystyle =\textstyle \sum _{3-3}'\sum _{3+3}\displaystyle \left({\frac {dx}{d{\text{x}}'}}{\frac {dx}{dx'}}{\frac {dy}{d{\text{y}}'}}{\frac {dy}{dy'}}{\frac {dz}{d{\text{z}}'}}{\frac {dz}{dz'}}\right)}$ ${\displaystyle =\textstyle \sum _{3+3}\displaystyle \left\{{\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\textstyle \sum _{3-3}'\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\right\}\quad }$ ${\displaystyle =\textstyle \sum _{3-3}\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\textstyle \sum _{3-3}'\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\cdot }$ (436)
Or, if we set
 ${\displaystyle H={\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}},}$ (437)
we shall have
 ${\displaystyle G=H^{2}.}$ (438)
It will be observed that ${\displaystyle F}$ represents the sum of the squares of the nine minors which can be formed from the determinant in (437), and that ${\displaystyle E}$ represents the sum of the squares of the nine constituents of the same determinant.

Now we know by the theory of equations that equation (431) will be satisfied in general by three different values of ${\displaystyle r^{2}}$, which we may denote by ${\displaystyle r_{1}^{2},r_{2}^{2},r_{3}^{2}}$, and which must represent the squares of the ratios of elongation for the three principal axes of strain; also that ${\displaystyle E,F,G}$ are symmetrical functions of ${\displaystyle r_{1}^{2},r_{2}^{2},r_{3}^{2}}$, viz.,

 ${\displaystyle E=r_{1}^{2}+r_{2}^{2}+r_{3}^{2},}$ ${\displaystyle F=r_{1}^{2}r_{2}^{2}+r_{2}^{2}r_{3}^{2}+r_{3}^{2}r_{1}^{2},}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}$ (439) ${\displaystyle G=r_{1}^{2}r_{2}^{2}r_{3}^{2}.}$

Hence, although it is possible to solve equation (431) by the use of trigonometrical functions, it will be more simple to regard ${\displaystyle \epsilon _{V'}}$ as a function of ${\displaystyle \eta _{V'}}$ and the quantities ${\displaystyle E,F,G}$ (or ${\displaystyle H}$), which we have expressed in terms of ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}\cdot }$ Since ${\displaystyle \epsilon _{V'}}$, is a single-valued function of ${\displaystyle \eta _{V'}}$ and ${\displaystyle r_{1}^{2},r_{2}^{2},r_{3}^{2}}$ (with respect to all the changes of which the body is capable), and a symmetrical function with respect to ${\displaystyle r_{1}^{2},r_{2}^{2},r_{3}^{2}}$, and since ${\displaystyle r_{1}^{2},r_{2}^{2},r_{3}^{2}}$ are collectively determined without ambiguity by the values of ${\displaystyle E,F,}$ and ${\displaystyle H}$, the quantity ${\displaystyle \epsilon _{V'}}$ must be a single-valued function of ${\displaystyle \eta _{V'},E,F,}$ and ${\displaystyle H}$. The determination of the fundamental equation for isotropic bodies is therefore reduced to the determination of this function, or (as appears from similar considerations) the determination of ${\displaystyle \psi _{V'}}$, as a function of ${\displaystyle t,E,F,}$ and ${\displaystyle H}$.

It appears from equations (439) that E represents the sum of the squares of the ratios of elongation for the principal axes of strain, that ${\displaystyle F}$ represents the sum of the squares of the ratios of enlargement for the three surfaces determined by these axes, and that ${\displaystyle G}$ represents the square of the ratio of enlargement of volume. Again, equation (432) shows that ${\displaystyle E}$ represents the sum of the squares of the ratios of elongation for lines parallel to ${\displaystyle X',Y',}$ and ${\displaystyle Z'}$; equation (434) shows that ${\displaystyle F}$ represents the sum of the squares of the ratios of enlargement for surfaces parallel to the planes ${\displaystyle {\text{X'-Y', Y'-Z', Z'-X'}}}$; and equation (438), like (439), shows that ${\displaystyle G}$ represents the square of the ratio of enlargement of volume. Since the position of the co-ordinate axes is arbitrary, it follows that the sum of the squares of the ratios of elongation or enlargement of three lines or surfaces which in the unstrained state are at right angles to one another, is otherwise independent of the direction of the lines or surfaces. Hence, ${\displaystyle {\tfrac {1}{3}}E}$ and ${\displaystyle {\tfrac {1}{3}}F}$ are the mean squares of the ratios of linear elongation and of superficial enlargement, for all possible directions in the unstrained solid.

There is not only a practical advantage in regarding the strain as determined by ${\displaystyle E,F,}$ and ${\displaystyle H}$, instead of ${\displaystyle E,F,}$ and ${\displaystyle G}$, because ${\displaystyle H}$ is more simply expressed in terms of ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$, but there is also a certain theoretical advantage on the side of ${\displaystyle E,F,H}$. If the systems of co-ordinate axes ${\displaystyle X,Y,Z}$, and ${\displaystyle X',Y',Z'}$, are either identical or such as are capable of superposition, which it will always be convenient to suppose, the determinant H will always have a positive value for any strain of which a body can be capable. But it is possible to give to ${\displaystyle x,y,z}$ such values as functions of ${\displaystyle x',y',z'}$ that ${\displaystyle H}$ shall have a negative value. For example, we may make

 ${\displaystyle x=x',}$⁠${\displaystyle y=y',}$⁠${\displaystyle z=-z'.}$ (440)
This will give ${\displaystyle H=-1}$, while
 ${\displaystyle x=x',}$⁠${\displaystyle y=y',}$⁠${\displaystyle z=z'.}$ (441)
will give #=1. Both (440) and (441) give ${\displaystyle H=1}$. Now although such a change in the position of the particles of a body as is represented by (440) cannot take place while the body remains solid, yet a method of representing strains may be considered incomplete, which confuses the cases represented by (440) and (441).

We may avoid all such confusion by using ${\displaystyle E,F,}$ and ${\displaystyle H}$ to represent a strain. Let us consider an element of the body strained which in the state ${\displaystyle (x',y',z')}$ is a cube with its edges parallel to the axes of ${\displaystyle X',Y',Z'}$, and call the edges ${\displaystyle dx',dy',dz'}$ according to the axes to which they are parallel, and consider the ends of the edges as positive for which the values of ${\displaystyle x',y',}$ or ${\displaystyle z'}$ are the greater. Whatever may be the nature of the parallelepiped in the state ${\displaystyle (x,y,z)}$ which corresponds to the cube ${\displaystyle dx',dy',dz'}$ and is determined by the quantities ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$, it may always be brought by continuous changes to the form of a cube and to a position in which the edges ${\displaystyle dx',dy'}$ shall be parallel to the axes of ${\displaystyle X}$ and ${\displaystyle Y}$, the positive ends of the edges toward the positive directions of the axes, and this may be done without giving the volume of the parallelepiped the value zero, and therefore without changing the sign of ${\displaystyle H}$. Now two cases are possible;—the positive end of the edge ${\displaystyle dz'}$ may be turned toward the positive or toward the negative direction of the axis of ${\displaystyle Z}$. In the first case, ${\displaystyle H}$ is evidently positive; in the second, negative. The determinant ${\displaystyle H}$ will therefore be positive or negative, we may say, if we choose, that the volume will be positive or negative, according as the element can or cannot be brought from the state ${\displaystyle (x,y,z)}$ to the state ${\displaystyle (x',y',z')}$ by continuous changes without giving its volume the value zero.

If we now recur to the consideration of the principal axes of strain and the principal ratios of elongation ${\displaystyle r_{1}^{2},r_{2}^{2},r_{3}^{2}}$ and denote by ${\displaystyle U_{1},U_{2},U_{3}}$ and ${\displaystyle U_{1}',U_{2}',U_{3}'}$ the principal axes of strain in the strained and unstrained element respectively, it is evident that the sign of ${\displaystyle r_{1},}$ for example, depends upon the direction in ${\displaystyle U_{1}}$ which we regard as corresponding to a given direction in ${\displaystyle U_{1}'}$. If we choose to associate directions in these axes so that ${\displaystyle r_{1},r_{2},r_{3}}$ shall all be positive, the positive or negative value of ${\displaystyle H}$ will determine whether the system of axes ${\displaystyle U_{1},U_{2},U_{3}}$ is or is not capable of superposition upon the system ${\displaystyle U_{1}',U_{2}',U_{3}'}$ so that corresponding directions in the axes shall coincide. Or, if we prefer to associate directions in the two systems of axes so that they shall be capable of superposition, corresponding directions coinciding, the positive or negative value of ${\displaystyle H}$ will determine whether an even or an odd number of the quantities ${\displaystyle r_{1},r_{2},r_{3}}$ are negative. In this case we may write

 ${\displaystyle r_{1}r_{2}r_{3}=H={\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}}.}$ (442)
It will be observed that to change the signs of two of the quantities ${\displaystyle r_{1},r_{2},r_{3}}$ is simply to give a certain rotation to the body without changing its state of strain.

Whichever supposition we make with respect to the axes ${\displaystyle U_{1},U_{2},U_{3}}$, it is evident that the state of strain is completely determined by the values ${\displaystyle E,F}$, and ${\displaystyle H}$, not only when we limit ourselves to the consideration of such strains as are consistent with the idea of solidity, but also when we regard any values of ${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$ as possible.

Approximative Formulæ.—For many purposes the value of ${\displaystyle \epsilon _{V'}}$ for an isotropic solid may be represented with sufficient accuracy by the formula

 ${\displaystyle \epsilon _{V'}=i'+e'E'+f'F+h'H,}$ (443)
where ${\displaystyle i',e',f',}$ and ${\displaystyle h'}$ denote functions of ${\displaystyle \eta _{V'}}$; or the value of ${\displaystyle \psi _{V'}}$ by the formula
 ${\displaystyle \psi _{V'}=i+eE'+fF+hH,}$ (444)

where ${\displaystyle i,e,f}$, and ${\displaystyle h}$ denote functions of ${\displaystyle t}$. Let us first consider the second of these formulæ. Since ${\displaystyle E,F}$, and ${\displaystyle H}$ are symmetrical functions of ${\displaystyle r_{1},r_{2},r_{3}}$, if ${\displaystyle \psi _{V'}}$ is any function of ${\displaystyle t,E,F,H}$, we must have

 ${\displaystyle {\frac {d\psi _{V'}}{dr_{1}}}={\frac {d\psi _{V'}}{dr_{2}}}={\frac {d\psi _{V'}}{dr_{3}}},}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (445) ${\displaystyle {\frac {d^{2}\psi _{V'}}{dr_{1}^{2}}}={\frac {d^{2}\psi _{V'}}{dr_{2}^{2}}}={\frac {d^{2}\psi _{V'}}{dr_{3}^{2}}},}$ ${\displaystyle {\frac {d^{2}\psi _{V'}}{dr_{1}dr_{2}}}={\frac {d^{2}\psi _{V'}}{dr_{2}dr_{3}}}={\frac {d^{2}\psi _{V'}}{dr_{3}dr_{1}}},}$

whenever ${\displaystyle r_{1}=r_{2}=r_{3}}$. Now ${\displaystyle i,e,f}$, and ${\displaystyle h}$ may be determined (as functions of ${\displaystyle t}$) so as to give to their proper values at every temperature for some isotropic state of strain, which may be determined by any desired condition. We shall suppose that they are determined so as to give the proper values to ${\displaystyle \psi _{V'}}$, etc., when the stresses in the solid vanish. If we denote by ${\displaystyle r_{0}}$ the common value of ${\displaystyle r_{1},r_{2},r_{3}}$ which will make the stresses vanish at any given temperature, and imagine the true value of ${\displaystyle \psi _{V'}}$