278 CAUCHY ON THE THEORY OF LIGHT. 
2 a Q 
ot+GtR=b: es 
by a diametral plane parallel to the given plane. The section 
thus obtained will be an ellipse, whose two axes will be numeri- 
cally equal to the velocities of propagation of the plane waves in 
the two rays. Furthermore, of these two rays, that one in which 
the plane waves propagate themselves with a velocity repre- 
sented by the major axis of the ellipse, will be polarized parallel 
to the minor axis; and reciprocally the ray in which the plane 
waves propagate themselves with a velocity represented by the 
minor axis of the ellipse, will be polarized parallel to the major 
axis. If we make the plane A, B, C, coincide with one cf the 
principal planes of the ellipsoid, the two polarized rays will follow 
the same course, and the two velocities of light in these rays will 
be precisely the velocities of propagation of the plane waves. 
Consequently, the velocities of light in the six polarized rays, 
whose directions coincide with the three axes of the ellipsoid, 
are two by two equal with one another, and to one of the num- 
bers / P, /Q, WR. Let us add, that the two rays, whose velo- 
city is / P, are polarized perpendicularly to the axis of # ; those 
whose velocity is  Q@, perpendicular to the axis of y, and those 
whose velocity is 4/ R perpendicular to the axis of z. In the 
particular case where the quantities P Q, become Ge to one 
another, the surface represented by the equation (10.), 0 
ae + 
Praia 9 oars 
becomes an ellipsoid of revolution, the axis of which is what we 
call the optical axis of the crystal. Then, one of the semi-axes 
of the section made by any diametral plane, is constantly equal 
to / Q, as well as the velocity of the light in one of the two 
polarized rays. The ray in question is that which we name 
“the ordinary ray,” and it is polarized parallel to the right line, 
which in the plane A B C forms the least and the greatest angle 
with the optical axis; whilst the other ray, called “ the extraor- 
dinary ray,” is polarized parallel to the right line of intersection 
of the plane A B C, and of a plane perpendicular to the optical 
axis. Also, the two rays, ordinary and extraordinary, are super- 
posed, when they are in the direction of the optical axis, and are 
reduced to a single ray, which no longer shows any trace of po- 
larization. 
