Page:Scientific Memoirs, Vol. 1 (1837).djvu/88

the two figures polarized perpendicularly to one another, we have generally,

${\begin{aligned}I_{o}&=\cos ^{2}b-\sin 2a\sin 2(a-b)\cos ^{2}\pi \left({\frac {o-e}{\lambda }}\right)\\I_{e}&=\sin ^{2}b+\sin 2a\sin 2(a-b)\sin ^{2}\pi \left({\frac {o-e}{\lambda }}\right),\end{aligned}}$

in which $\lambda$ indicates the length of undulation for a definite colour, $o-e$ the difference of path of both rays, and I the intensity of the polarized light falling perpendicularly upon the crystal plate. Now if the axis of the plate is made to form an angle of 45° with the plane of primitive polarization, that is to say, if we suppose $a=45^{\circ }$, we shall have,

${\begin{aligned}I_{o}&=\cos ^{2}b-\cos 2b\cos ^{2}\pi \left({\frac {o-e}{\lambda }}\right)\\I_{e}&=\sin ^{2}b+\cos 2b\sin ^{2}\pi \left({\frac {o-e}{\lambda }}\right).\end{aligned}}$

If then by any means we can make the difference between the paths of both rays equal to an uneven number of quarter-undulations, the second condition will also be satisfied as well as the first, viz. that of the equal intensity. Suppose, for instance.

$o-e=\left[\left({\frac {2n-1}{4}}\right)\lambda \right],$,

then will

${\begin{aligned}I_{0}&=\cos ^{2}b-{\frac {1}{2}}\cos 2b={\frac {1}{2}}\\I_{e}&=\sin ^{2}b+{\frac {1}{2}}\cos 2b={\frac {1}{2}}.\end{aligned}}$

The difference of path $o-e$ depends on two quantities; on the thickness of the plate, to which it is in direct proportion, and on the difference of velocity of the two rays which pass through the plate, that is to say, on the constant of double refraction/

Airy's method consists only in varying the thickness of the plate by splitting it, whilst the double refraction remains the same, until the difference between the paths of the rays is equal to an uneven number of quarter-undulations. As biaxal mica under a perpendicular incidence of the light is similar to an uniaxal crystal and best allows splitting into larger plates, its application will therefore be preferable. I, on the contrary, alter the double refraction of the substance, whilst the thickness remains the same, until the required difference of path is obtained.

To alter the refraction of rays in a crystallized lamina by pressure or change of temperature, so that it may exhibit the desired effect in a given thickness. would afford no convenient practical arrangement. It is, however, very easy by means of pressure or cooling to change the uncrystallized into a double-refracting body, which gives precisely the required effect. In the apparatus proposed by Fresnel, consisting of