Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/209

In crystals of the orthorhombic system, the three ellipsoids will have their axes parallel to the rectangular crystallographic axes. If we take these directions for the axes of coordinates, ${\text{E, F, G,}}{\text{E}}',{\text{F}}',{\text{G}}',e,f,g$ will vanish and equation (13) will reduce to

${\text{V}}^{2}={\frac {a\alpha ^{2}+b\beta ^{2}+c\gamma ^{2}}{\rho ^{2}}}\cdot$

If the coordinate axes are so placed that

$a>b>c,$

the optic axes will lie in the X-Z plane, making equal angles $\phi$ with the axis of Z, which may be determined by the equation

$\tan ^{2}\phi ={\frac {a-b}{b-c}}={\frac {p^{2}({\text{A}}-{\text{B}})-4\pi ^{2}({\text{A}}'-{\text{B}}')}{p^{2}({\text{B}}-{\text{C}})-4\pi ^{2}({\text{B}}'-{\text{C}}')}}\cdot$

To get a rough idea of the manner in which $\phi$ varies with the period, we may regard ${\text{A, B, C,}}{\text{A}}',{\text{B}}',{\text{C}}',$ as constant in this equation.

But since the lengths of the axes of the ellipsoid ($a,b,$ etc.) vary with the period, it may easily happen that the order of the axes with respect to magnitude is not the same for all colors. In that case, the optic axes for certain colors will lie in one of the principal planes, and for other colors in another. For the color at which the change takes place, the two optic axes will coincide. The differential coefficient ${\frac {d\phi }{dp}}$ becomes infinitely great as the optic axes approach coincidence.

In crystals of the monodinic system, each of the three ellipsoids will have an axis perpendicular to the plane of symmetry. We may choose this direction for the axis of X. Then $>{\text{F, G,}}{\text{F}}',{\text{G}}',f,g$ will vanish and equation (13) will reduce to

${\text{V}}^{2}={\frac {a\alpha ^{2}+b\beta ^{2}+c\gamma ^{2}+e\beta \gamma }{\rho ^{2}}}\cdot$

The angle $\theta$ made by one of the axes of the ellipsoid ($a,b,$ etc.) in the plane of symmetry with the axis of Y and measured toward the axis of Z, is determined by the equation

$\tan 2\theta ={\frac {e}{c-b}}={\frac {p^{2}{\text{E}}-4\pi ^{2}{\text{E}}'}{p^{2}({\text{C}}-{\text{B}})-4\pi ^{2}({\text{C}}'-{\text{B}}')}}\cdot$

To get a rough idea of the dispersion of the axes of the ellipsoid ($a,b,$ etc) in the plane of symmetry, we may regard ${\text{B, C, E, }}{\text{B}}',{\text{C}}',{\text{E}}',$ as constant in this equation, and suppose the axis of Y so placed as to make ${\text{E}}$ vanish.

It is evident that in this system the plane of the optic axes will be fixed, or will rotate about one of the lines which bisect the angles made by the optic axes, according as the mean axis of the ellipsoid