Page:Scientific Papers of Josiah Willard Gibbs.djvu/247

the positive or toward the negative direction of the axis of $Z$. In the first case, $H$ is evidently positive; in the second, negative. The determinant $H$ 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 $(x,y,z)$ to the state $(x',y',z')$ 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 $r_{1}^{2},r_{2}^{2},r_{3}^{2}$ and denote by $U_{1},U_{2},U_{3}$ and $U_{1}',U_{2}',U_{3}'$ the principal axes of strain in the strained and unstrained element respectively, it is evident that the sign of $r_{1},$ for example, depends upon the direction in $U_{1}$ which we regard as corresponding to a given direction in $U_{1}'$. If we choose to associate directions in these axes so that $r_{1},r_{2},r_{3}$ shall all be positive, the positive or negative value of $H$ will determine whether the system of axes $U_{1},U_{2},U_{3}$ is or is not capable of superposition upon the system $U_{1}',U_{2}',U_{3}'$ 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 $H$ will determine whether an even or an odd number of the quantities $r_{1},r_{2},r_{3}$ are negative. In this case we may write

$r_{1}r_{2}r_{3}=H={\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}}.$

(442)

It will be observed that to change the signs of two of the quantities $r_{1},r_{2},r_{3}$ 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 $U_{1},U_{2},U_{3}$, it is evident that the state of strain is completely determined by the values $E,F$, and $H$, 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 ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ as possible.

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

$\epsilon _{V'}=i'+e'E'+f'F+h'H,$

(443)

where $i',e',f',$ and $h'$ denote functions of $\eta _{V'}$; or the value of $\psi _{V'}$ by the formula

$\psi _{V'}=i+eE'+fF+hH,$

(444)