Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/224

In the case of uniaxial crystals, the direction of the optic axis is fixed. We may therefore write

$\theta =n_{\text{R}}^{2}n_{\text{L}}^{2}\left({\frac {\text{K}}{\lambda ^{2}}}+{\frac {{\text{K}}'}{\lambda ^{4}}}\right),$

(40)

regarding ${\text{K}}$ and ${\text{K}}'$ as constants. If we had used equation (37), we should have had the factor $n^{4}$ instead of $n_{\text{R}}^{2}n_{\text{L}}^{2}.$ Since this factor varies but slowly with $\lambda ,$ it may be neglected, if its omission is compensated in the values of ${\text{K}}$ and ${\text{K}}'.$ The formula being only approximative, such a simplification will not necessarily render it less accurate.

20. But without any such assumption as that contained in the last paragraph, we may easily obtain formulæ for the experimental determination of $\Phi$ and $\Phi '$ for the optic axis of a uniaxial crystal. Considerations analogous to those of § 13 of the former paper (page 190 of this volume), show that in differentiating equation (39) we may regard $\Phi$ and $\Phi '$ as constant, although they may actually vary with $\lambda .$ This equation may be written

${\frac {\theta \lambda ^{2}}{n^{4}}}={\frac {\Phi }{2k^{2}}}-{\frac {2\pi ^{2}\Phi '}{\lambda ^{2}}}\cdot$

(41)

Therefore,

${\frac {d\left({\frac {\theta \lambda ^{2}}{n^{4}}}\right)}{d\left({\frac {1}{\lambda ^{2}}}\right)}}=-2\pi ^{2}\theta '.$

(42)

When $\Phi '$ has been determined by this equation, $\Phi$ may be found from the preceding.

21. If we wish to represent $\phi$ geometrically, like ${\text{U}}_{1}$ and ${\text{U}}_{2},$ we may construct the surfaces

$\scriptstyle {\text{A}}\displaystyle x^{2}+\scriptstyle {\text{B}}\displaystyle y^{2}+\scriptstyle {\text{C}}\displaystyle z^{2}+\scriptstyle {\text{E}}\displaystyle yz+\scriptstyle {\text{F}}\displaystyle zx++\scriptstyle {\text{G}}\displaystyle xy=\pm 1,$

(43)

the coefficients $\scriptstyle {\text{A, B,}}$ etc., being the same by which $\phi$ is expressed in terms of ${\text{L}}^{2},{\text{M}}^{2},$ etc. The numerical value of $\phi ,$ for any direction of the wave-normal, will thus be represented by the square of the reciprocal of the radius vector of the surface drawn in the same direction. The positive or negative character of $\phi$ must be separately indicated. There are here two cases to be distinguished. If the sign of is the same in all directions, the surface will be an ellipsoid, and we have only to know whether all the values of $\phi$ are to be taken positively or all negatively. But if $\phi$ is positive for some directions and negative for others, the surface will consist of two conjugate hyperboloids, to one of which the positive, and to the other the negative values belong.