DE. PLTJCKEE ON A NEW GEOMETEY OF SPACE. 
763 
Accordingly the plane of refraction, conjugate to (6), is represented by the equation 
dE dE 
dxQ- dy$ > ’ > 
( 12 ) 
which may be expanded into the following one, 
(Ax+By+T>z)q=(Bx+Cy+'Ez)p, (13) 
or 
{Aq— Bp)x-\-(¥>q— Cp)y+(Dq— Ej?)z=0* (14) 
8. These equations remain unaltered if p and q vary in such a way that the ratio ^ 
remains the same, i. e. if the angle of incidence vary while the plane , of incidence 
remains the same. The same equations do not contain w, the value of which depends 
upon the density of the surrounding medium. Hence 
All rays of light confined within the same plane of incidence, after being divided into 
two by double refraction , are confined again within the same plane — the plane of refrac- 
tion. This plane remains the same if the surrounding medium be changed. 
9. The plane xy, i. e. the surface of the crystal, containing the trace (11), its conju- 
gate diameter, the equations of which are 
or 
«-0 
(15) 
Atf+B.y+D^O, | 
B#+Qy+E;z =0, j 
(16) 
is confined within the plane of refraction, whatever may be the incident ray. The same 
may be proved analytically by observing that (12) is satisfied by means of the two equa- 
tions (15). Hence 
A ray of light of any direction whatever meeting the surface of a biaxal crystal in a 
fixed point is so refracted that the plane containing both refracted rays passes through a 
fixed right line (15). 
* On representing any one of both refracted rays by the equations 
x=rz, y=sz, 
the last equation, written thus, 
(A 2 -B i >>+(B 2 -Cp) S .+(D ? -Ep)=0, (1) 
indicates a relation between the direction of the incident ray, determined by the constants p and q, and the 
direction of the refracted one, determined by r and s. 
This equation will not be altered if the incident ray, moved parallel to itself, meet the section of the crystal 
in any point 
x= ?> y=r. 
If r and s be regarded as variable, and <r being constant, the equation (1) represents the plane of refraction 
corresponding to the incident ray 
x=pz-\-§, y=qz+c, 
and containing both refracted rays. 
