68 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
BIOT'S OR FRESNEL'S RULE. 
In several of the optical methods to be described frequent use is made of 
a rule first formulated by Biot,* by which the directions of extinction for 
any section of a birefracting mineral can be found. Some ten years before 
Biot announced this general rule, Malusf had found that the light-waves 
emerging from a calcite rhomb were plane polarized and that for any given 
section of calcite the lines of extinction were parallel with and at right angles 
to the trace of the plane containing the optic axis and the normal to the 
section ; in other words, the orthogonal projection of the optic axis on any 
given section of a uniaxial mineral determines its lines of extinction, which 
are parallel with and normal to this projection line. By modifying the 
wording of this rule slightly, it is possible, as Biot proved experimentally 
and FresnelJ demonstrated theoretically, to make it of general application 
to all birefracting substances; thus, the directions of extinction of a biaxial 
mineral^ cut after any section are parallel to the traces (on that section) of 
the planes bisecting the angles between the two planes containing the normal 
to the section and the optic axes (optic binomials) ; in other words, the lines 
bisecting the angles between the lines of orthogonal projection of the optic 
binormals on any given section of a biaxial mineral are the directions of 
extinction for that section for the particular color of light employed. It 
should be noted that this rule applies to optic phenomena within the crystal 
itself, and that for oblique incidence of light, as in convergent polarized 
light, the angles of incidence must be reduced to the angles which the 
refracted waves include with the normal to the plate. This is usually 
accomplished by means of the formula sin V = , E being the observed 
/3 
angle, V the required angle, and /3 the intermediate refractive index of the 
mineral. Strictly speaking, it is not correct to use the refractive index /3 
in this equation, but in minerals of weak to medium birefringence the 
difference between /3 and the correct index is so slight that the error intro- 
duced by using /3 is practically negligible. The effect of the boundary 
surfaces in rotating the plane of polarization of transmitted waves is also, 
in general, slight for small angles of incidence and is usually disregarded in 
practical work. 
*Biot, J. B.. Mem. de 1'Acad. de I'lnst. de France, 3, 238. 1820. 
tMalus, E. L.. TWorie de la double refraction de la lumiere dans les substances cristallisles. Mem. pro's. 
A Tlnst. Sc. math. et. phys., 2. 303. 1811. 
IFresnel. Second Mcmoirc sur la double refraction, Fogg. Am.. 23. 542. 1831. 
IThe uniaxial minerals may for the moment be considered the limiting case of biaxial minerals for 
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