Page:Scientific Papers of Josiah Willard Gibbs.djvu/245

Or, if we set

$H={\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}},$

(437)

we shall have

$G=H^{2}.$

(438)

It will be observed that $F$ represents the sum of the squares of the nine minors which can be formed from the determinant in (437), and that $E$ 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 $r^{2}$, which we may denote by $r_{1}^{2},r_{2}^{2},r_{3}^{2}$, and which must represent the squares of the ratios of elongation for the three principal axes of strain; also that $E,F,G$ are symmetrical functions of $r_{1}^{2},r_{2}^{2},r_{3}^{2}$, viz.,

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

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

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 $F$ represents the sum of the squares of the ratios of enlargement for the three surfaces determined by these axes, and that $G$ represents the square of the ratio of enlargement of volume. Again, equation (432) shows that $E$ represents the sum of the squares of the ratios of elongation for lines parallel to $X',Y',$ and $Z'$; equation (434) shows that $F$ represents the sum of the squares of the ratios of enlargement for surfaces parallel to the planes ${\text{X'-Y', Y'-Z', Z'-X'}}$; and equation (438), like (439), shows that $G$ represents the square of the ratio of