of the Vibrations of Polarized Light by Diffraction. 325 
observed point from the origin of coordinates, we have 
p= Vx? + (y—B)?+ (z—y)?=p—mB—ny, 
where 
p= Varty+2, m= 
and /, m, and v being the cosines of the angles made by the 
diffracted ray with the coordinate axes, /?+m?+n?=1. Now 
from (9) we get 
(wt, y, 2) =4ak& sin k (wi—by—cz), 
O=akéS, 
whence 
where 
nt Sa f dB {dy sin k[wt—p + (m—B)8 + (n—e)y]. 
And the values of the functions that enter into (5) being found 
in a similar manner, we get 
uy ga 1 (1E +mn +n8)]S, 
v,=$k(a+l) [n—m(lE+ mn +n8)]8, 
=ghk(a+l)[(€—n(lE+ mn +né) |S. 
These te a hold good also for the waves reflected from 
the opening, only that in this case / is negative. For a point in 
the direction opposite to that of the meident ray a+/=0, and 
therefore all the components of the motion are equal to nothing. 
Mr. Stokes has arrived at the same result, although he did 
not regard the reflected wave, and has not completely solved the 
problem. Ifa plane be supposed to pass through the refracted 
and the incident ray, and if « denote the angle which the vibra- 
tions of the incident ray make with the normal to this plane, «, 
the angle made by the vibrations of the diffracted ray with the 
same normal, and 8 the angle of diffraction, then we easily find, 
as Stokes has done, that 
tan a, = cos P tan a, 
which is independent of the form and position of the opening. 
The vibrations therefore become more nearly vertical after pass- 
ing through a vertical slit or grating. Accordingly, therefore, 
as experiment shows that the plane of polarization is rendered 
more vertical or more horizontal by diffraction, so must the vibra- 
tions of polarized light be parallel or perpendicular to the plane 
of polarization. It must, however, be renembered that, mathe- 
matically speaking, the screen is supposed to be a plane which 
does not itself vibrate, and which reflects no light from its edges. | 
