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of Mr. Mac Cullagh’s briefer paper, which have appeared to me to 
deserve a peculiar and special attention. I mean the geometrical 
elegance of the principal enunciations, and the philosophical cha- 
racter of the interspersed remarks. 
As a specimen of the former, I shall select the theorem of the 
polar plane. When light in air is incident on a doubly-refracting 
crystal, it may be polarised in such a plane, that one of the two 
refracted rays'shall disappear ; and then the one refracted vibration 
which corresponds to the one remaining refracted ray, must (by 
the hypotheses or laws already mentioned) be the resultant of the 
one incident and one reflected vibration ; and consequently these 
three vibrations must be contained in one common plane, which 
plane it is therefore an object of interest to assign a simple rule 
for constructing. In fact, the refracted vibration is known, in 
direction, from the laws of propagation of light in the crystal, and 
the hypotheses already mentioned ; if, then, we know how to draw 
through its direction the plane just now referred to, we should only 
have to examine in what lines this plane intersected the incident 
and reflected waves, in order to obtain the direction of the incident 
and reflected vibrations, and afterwards (by the rules of statical 
composition) the relative magnitudes of all the three vibrations, or 
the relative intensities of the incident, reflected, and refracted lights. 
Now Mr. Mac Cullagh shows, that the desired construction can be 
deduced from the properties of the doubly refracting medium or 
wave, as follows: Let or, op represent in length and in direction 
the velocity of the refracted ray, and the slowness of the re- 
fracted wave; so that, by what has been before supposed, the 
refracted vibration ov is perpendicular to the plane Tor ; then, 
if a plane be drawn through the vibration ov, parallel to the 
line tp, this plane, which Mr. Mac Cullagh calls the polar plane 
of the ray or, will be the plane desired; that is, it will contain the 
incident and the reflected vibrations, if these be uniradial, or, in 
other words, if they have such directions, or correspond to such 
polarisations, as to cause one of the two refracted rays in the crystal 
to disappear. 
Many elegant geometrical corollaries are drawn, in the Essay, 
from this theorem of the polar plane ; but I shall only mention one, 
