DR. PLUCKER ON A NEW GEOMETRY OF SPACE. 
765 
into which any incident ray is divided , and that fixed direction , are confined within the 
same plane. 
12. By putting 
Ey=Ey, 
the equation of the plane of refraction becomes 
(Aq—~Bp)x+ (B<?- Op>= 0, 
which, after eliminating p and q, may be written thus, 
(AE-DB>+(BE-DC)y=0 (19) 
In this case the plane of refraction is perpendicular to xy and passes through OZ. 
The plane of incidence perpendicular to xy, or its trace within this plane, is represented 
by 
I (20) 
It is easily seen that this trace is perpendicular to the trace of that diametral plane 
which, with regard to the ellipsoid E, is conjugate to OZ. Indeed this plane is repre- 
sented by 
“=»H-%+F*=0, 
and its trace within xy by 
Rr-f Ey=0. 
Each ray within the plane of incidence (20) is divided by double refraction into two, 
both confined within the same vertical plane of refraction. That is especially the case 
with regard to the ray incident at right angles ; the corresponding plane of refraction, 
represented by (19), contains the incident ray and both the refracted rays. 
13. Besides the vertical ray, there is in each plane of incidence one ray confined with 
both refracted rays within the same plane. After eliminating p and q between the 
general equations of the planes of incidence and of refraction, 
qx—py, 
(Ax + By + ~Dz)q=(Bx +Cy+ E z)p, 
the following equation is obtained, 
B(y—x*)+(A-C)xy + (-Dy-Ex)z=0, (21) 
representing a cone of the second degree, the locus of incident rays which are confined 
within their corresponding planes of refraction. This cone passes through the vertical 
OZ, and intersects xy within two right lines perpendicular to each other. These lines 
are congruent with the two axes of the ellipse 
Aa- 2 +2B^4-Oy 2 =l, (22) 
along which the plane xy is intersected by the ellipsoid E. (That is instantly seen by 
putting B=0 [4].) Hence both rays, grazing the surface of the crystal along the axes 
of the ellipse (22), are confined with both corresponding refracted rays within the same 
plane. 
MDCCCLXY. 5 N 
