29 
lem is, therefore, completely solved in this case; and it is 
obvious, that a construction precisely similar will apply to the 
other case, in which or’ is the only refracted ray. The plane 
B, which, in this second case, answers to the plane a in the 
first case, is perpendicular to the plane o1’m, and parallel 
to the right line Tm. 
If the incident ray be polarized in a direction interme- 
diate between the two transversal directions which give only 
a single refracted ray, the incident vibration may be resolved 
into two vibrations parallel to those two transversals. ‘The 
reflected vibrations arising from each of the component 
incident vibrations are to be found by the foregoing rules, 
and then to be compounded. 
When the intersection of the planes a and Bis perpendi- 
cular to the direction of the reflected ray, this ray is polar- 
ized parallel to that intersection, whatever be the plane of 
polarization of the incident ray. The angle of incidence at 
which this takes place is the polarizing angle. 
When the refracted ray oT or o1’ is a normal to the wave- 
surface, the plane a or B is the plane of polarization of the 
ray. For example, if or be the ordinary ray in a uniaxal 
crystal, the plane a contains the ray or and the axis of the 
crystal, 
The hypotheses from which Mr. Mac Cullagh has obtained 
the foregoing laws, are these :-— 
1. The density of the ether is the same in all media. 
2. The vibrations are parallel to the plane of polarization. 
3. The vis viva is preserved. 
4. The vibrations are preserved: that is, the resultant of 
the incident and reflected vibrations is the same as the resul- 
tant of the refracted vibrations. 
The author finds that his theory represents very accu- 
rately the experiments of Sir David Brewster and M. Seebeck, 
on the light reflected in air from a surface of Iceland spar. 
