760 
DE. PLUCKEE ON A NEW GrEOMETEY OF SPACE. 
II. — On Complexes of Luminous Rays within Biaxal Crystals. 
1. A single ray of light when meeting the surface of a doubly refracting crystal is 
divided into two rays determined by means of their four coordinates, r, s, g, a. All inci- 
dent rays constituting a configuration, especially all rays starting from a luminous point 
and forming a conical surface, constitute within the crystal a new configuration, repre- 
sented by the system of three equations between ray-coordinates. All incident rays 
constituting a congruency, emanating, for instance, in all directions from a luminous 
point, constitute within the crystal, after refraction, another congruency. Finally, a 
complex of incident rays, all rays, for instance, emanating in all directions from every 
point of a luminous curve, constitute within the crystal another complex of refracted 
rays. The congruency of refracted rays is represented by two, the complex by a single 
equation between ray-coordinates. 
2. But before entering into the discussions indicated by the foregoing remarks, a 
short digression on double refraction might be desirable. 
A biaxal crystal being cut along any plane whatever, we may suppose that this plane 
is congruent with xy , and that the point where an incident ray meets it is the origin of 
coordinates O. Let /n x 
x=pz, y=qz (1) 
be the equations of the incident ray, whence 
(2) 
P 9 
the equation of the plane of incidence. In the moment of Incidence the front of the 
corresponding elementary wave, perpendicular to the ray, will be represented by 
z+qy+px = 0 (3) 
After the front of the wave has moved in air through the unit of distance, its equation 
becomes , ... 
z+qy+px=w (4) 
on putting 
At this moment the front of the wave intersects xy along a right line, which we may 
denote by HR, the equation of which is 
qy+px=w (5) 
If the optical density of the surrounding medium increases, the value of w decreases 
in the same ratio. 
3. Around the point O, where the incident ray meets the section of the crystal, let 
the wave-surface be described as it is at that moment when the front of the elementary 
wave intersects xy along RR. The position of the axes of elasticity of the crystallized 
medium being known with regard to the axes of coordinates, the equation of the wave- 
surface only depends upon three constants a, #, c, which are to be referred to the same 
