Page:Scientific Papers of Josiah Willard Gibbs.djvu/221

the state of reference, and $x,y,z$, 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 $x,y,z$ as functions of $x',y',z'$, the form of the functions determining the second state of strain. For brevity, we may sometimes distinguish the variable state, to which $x,y,z$ relate, and the constant state (state of reference) to which $x',y',z'$ 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 $x,y,z$ and $x',y',z'$, 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 $x,y,z$ and $x',y',z'$ relate will be distinguished as the axes of $X,Y,Z$ and $X',Y',Z'$. 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 $x,y,$ and $z$ with respect to $x',y',$, and $z'$; for changes in the values of $x,y,z$, 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.,

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

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 $\epsilon _{V}$ and $\eta _{V}$ 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

$\delta \epsilon _{V}=t\delta \eta _{V}.$