Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/192

The variational equation of motion is

$\iiint \left\{\rho _{1}{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}\delta e_{x}+\rho _{1}{\frac {\partial ^{2}e_{y}}{\partial t^{2}}}\delta e_{y}+\rho _{1}{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}\delta e_{z}\right\}\ dx\ dy\ dz$                                 

$=-\iiint \left\{\delta \phi -p\delta \left({\frac {\partial e_{x}}{\partial x}}+{\frac {\partial e_{y}}{\partial y}}+{\frac {\partial e_{z}}{\partial z}}\right)\right\}\ dx\ dy\ dz$

where p denotes an undetermined function of (x, y, z): the term in p being introduced on account of the kinematical constraint expressed by the equation

${\text{div }}\mathbf {e} =0$

The equations of motion which result from this variational equation are

$\rho _{1}{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=-{\frac {\partial p}{\partial x}}+n\nabla ^{2}e_{x}$

and two similar equations. It is evident that p resembles a hydrostatic pressure.

Substituting in these equations the analytical expression for a plane wave, we readily find that the velocity V of the wave is connected with the direction-cosines (λ, μ, ν) of its normal by the equation

${\frac {\lambda ^{2}}{n-\rho _{1}V^{2}}}+{\frac {\mu ^{2}}{n-\rho _{2}V^{2}}}+{\frac {\nu ^{2}}{n-\rho _{3}V^{2}}}=0$

When this is compared with Fresnel's relation between the velocity and direction of a wave, it is seen that the new formula differs from his only in having the reciprocal of the velocity in place of the velocity. About 1867 Stokes carried out a series of experiments in order to determine which of the two theories was most nearly conformable to the facts: he found the construction of Huygens and Fresnel to be decidedly the more correct, the difference between the results of it and the rival construction being about 100 times the probable error of observation.^[1]

1. Proc. R. S., June, 1872. After these experiments Stokes gave it as his opinion (Phil. Mag. xli (1871), p. 521) that the true theory of crystal-optics was yet to be found. On the accuracy of Fresnel's construction of. Glazebrook, Phil. Trans. clxxi (1879) p. 421, and Hastings, Am. Journ. Sci. (3) xxxv (1887) p. 60.