487 
that in Mac Cullagh’s theory the direction of vibration in an 
extraordinary medium is perpendicular to the plane of the ray 
and wave normal, and that the polar plane of a refracted ray 
is by definition a plane through the point of incidence parallel 
to the direction of vibration, and also parallel to a line joining 
the extremity of the ray with the corresponding point on the 
Index surface,—the last-mentioned surface being the polar re- 
ciprocal of the refracted wave-surface, taken with respect to 
the reflected wave-surface, or wave-sphere, contemporaneously 
generated. We have to consider a ray of polarized light in- 
cident on the surface of an extraordinary medium, and giving 
rise to a reflected ray and a single refracted ray. Let the in- 
cident ray and the reflected ray be respectively produced 
within the medium, and let the three rays, viz., the incident 
ray produced, the refracted ray, and the reflected ray pro- 
duced, be represented in direction (see Fig. 2) by Ak, AR’ and 
AR"; and take AR=AR"=1 
as the radius of the wave-sphere 
and Af’ as the radius of the 
wave-surface, corresponding at 
a given instant of time to the 
first or ordinary medium and 
the extraordinary medium re- 
spectively. Take also AW as 
the perpendicular on the tan- 
gent plane of the wave-surface : Fig. 2. 
at R’, or ‘wave-normal,’ corresponding to the refracted ray 
AR’; and let AN represent the normal to the plane of sepa- 
ration of the two media, and AH the intersection of the last- 
mentioned plane with the plane of incidence. The lines 4R 
Ak", AW, AN, AH are of course all of them in the plane of 
incidence, the line AWN bisects the angle made by the lines 
AR, AR", and the lines AN, AH are at right angles to each 
other. The length of the wave-normal A W is given by the 
equation AR sin NAW= AW sin NAR, or putting, as above, 
