76 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
through EO and the line of propagation of the incident wave or, in short, 
of finding the point of intersection F of the line of propagation EF with the 
refracted wave-front FON. On emergence from the plate the same process 
is repeated and the direction OF is projected back to the incident wave- 
front by the line FE' normal to the refracted wave-front. From the spheri- 
cal triangle PQR, (Fig. 48) the relation is readily obtained 
cos (* r) =cot C tan e, or cot =cos ( r) cot e 
Similarly, for the azimuth of the line of vibration of the emergent wave we 
have 
cot ' = cos (i r) . cot C =cos 2 (* r) . cot e 
This is the fundamental formula of Fresnel for waves transmitted through 
an isotropic plate. The angle Z may be taken either between the line of 
vibration and the normal to the plane of incidence or between the plane of 
polarization (normal to the plane of vibration) and the plane of incidence. 
The above equation indicates that the amount of rotation (C' = e) of the 
plane of polarization increases with the difference (' r) or with the inclina- 
tion of the incident ray and also with the azimuth e. For e = o or 90, C = o 
and no rotation occurs ; for a given angle of incidence the amount of rota- 
tion is greatest for e = 45 in which case 
cot C' = cos 2 (i-r) 
From the above it is evident that at each lens surface the planes of polar- 
ization of all transmitted waves are in general rotated slightly, the amount 
of rotation increasing with the angle of incidence and ranging up to 5 and 
more for steeply inclined rays. This rotation occurs at all surfaces of the 
condenser lenses, of the glassmount and of the objective lenses, and gives rise 
to the faint dark cross observed when viewing the image C"D" (Fig. 30) 
in the rear focus of the objective, even when no mineral plate is under the 
objective.* This cross is accentuated if the ordinary type of nicol prism 
be used with oblique end surfaces, in which case the beam of polarized 
waves emerging from the polarizer itself is rotated slightly. If, moreover, 
the lenses of the condenser or objective systems are not carefully mounted, 
their brass supports may produce strain, which in turn affects the polarized 
waves and illuminates the field to a greater or less extent. 
The crystal plate, from which the interference figure is obtained, also has 
a rotatory effect on the plane of polarization of the transmitted rays and 
tends to decrease the accuracy of any measurement which depends on the 
degree of curvature or the exact location of points on the black bars or zero 
isogyres of the interference figure. A second factor which tends to diminish 
the accuracy of measurements of interference figures is the objective itself. 
An oblique parallel beam of light is unfortunately not brought to focus at a 
point as indicated in conventional diagrams of interference figures, but to 
two different lines, the one vertical and the second horizontal; as these 
foci are not located in the same plane, the result is that the horizontal lines 
The first correct explanation of this dark cross was given apparently by F.Rinnr. Centralblatt fur Miner.. 
1900, 88-89; G. Cesaro. Bull, de 1'Acad. roy. de Belgique (Classe des Sciences, 1006. 459) has also described 
the phenomenon and directed attention to the fact that from Fresncl's formula for refraction it is evident 
that the plane of vibration is not changed on the passage of a plane polarized light-wave from one isotropic 
medium into a second. Compare also P. E. Wright. Aracr. Jour. Set. (4). 31, 187. 1911. 
