Page:Scientific Papers of Josiah Willard Gibbs.djvu/231

surface perpendicular to it is as great as for any other surface passing through the axis of $X$. Then, if we write ${\frac {dS}{d\alpha }}$, ${\frac {dS}{d\beta }}$, ${\frac {dS}{d\gamma }}$ for the differential coefficients derived from the last equation by treating $\alpha ,\beta$ and $\gamma$ as independent variables,

${\frac {Ds}{d\alpha }}d\alpha +{\frac {dS}{d\beta }}d\beta +{\frac {dS}{d\gamma }}d\gamma =0,$

when	$\alpha d\alpha +\beta d\beta +\gamma d\gamma =0,$
and	$\alpha =1,$$\beta =0,$$\gamma =0,$
That is,	${\frac {dS}{d\beta }}=0,$and${\frac {dS}{d\gamma }}=0,$
when	$\alpha =1,$$\beta =0,$$\gamma =0.$
Hence	$X_{Y}=0,$and$Z_{X}=0.$	(390)
Moreover,	${\frac {dS}{d\beta }}d\beta +{\frac {dS}{d\gamma }}d\gamma =0,$
when	$\alpha =0,$$d\alpha =0,$
	$\beta d\beta +\gamma d\gamma =0,$
and	$\beta =1,$$\gamma =0.$
Hence	$Y_{Z}=0.$	(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 ($\alpha ,\beta ,\gamma$) are by (379)

$\alpha X_{X},$ $\beta Y_{T},$ $\gamma Z_{Z}.$

(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 $X_{X},Y_{Y},Z_{Z}$ 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