764 
DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
If without the crystal the ‘plane of incidence turns round the perpendicular to the 
section , within the crystal the plane of refraction simultaneously turns round the diameter 
of the third ellipsoid conjugate to the section. 
10. In order to construct the plane of refraction, we want to know another diameter 
conjugate to any plane passing through the trace (11). In selecting among these planes 
the wave-front itself in its primitive position, the plane of refraction will be obtained by 
drawing a plane through both diameters conjugate to the section of the crystal and the 
primitive wave-front. 
The wave-front in its primitive position is represented by 
px+qy+z= 0, 
its conjugate diameter by the equations 
rfB dE j 
^ dz ^ I 
[ (17) 
dy ^ " dz ’ j 
which, if expanded, become 
Ax + By + Dz =p( Dx+ E y -j- F 2 ), | 
Dx-\-Oy-\-Dz = g{Dx-\-Dy-\-Dz).\ 
In order to prove in the analytical way that the diameter conjugate to the primitive 
wave-front falls within the plane of refraction, it is sufficient to observe that, by elimi- 
rfE 
nating -gp- between the two equations (17), the equation of the plane of refraction (12) 
is obtained. 
11. If a ray of light meet the surface of a crystal in a given point, the third ellipsoid 
remains invariably the same as long as the position of the crystal is not altered. There- 
fore the diameter conjugate to the wave-front remaining likewise the same, whatever 
may be the section of the crystal passing through the point of incidence, the plane of 
refraction always passes through that fixed diameter. Again, if the incident ray, dis- 
placed parallel to itself, meet the surface of the crystal in a new point, this new point of 
incidence becomes the centre of the third ellipsoid, likewise displaced parallel to itself. 
The diameter conjugate to the primitive wave-front, always passing through the point 
of incidence, retains the same direction. We may finally observe that the surface of the 
crystal, if a curved one, may be replaced for any incident ray by the plane tangent to it 
in the point of incidence. 
If a ray of light meet a biaxal crystal in a given point , whatever may be the surface 
bounding the crystal and containing that point, the plane of refraction passes through a 
fixed right line. 
If a system of parallel rays meet the surface of a biaxal crystal , each ray of which 
after double refraction is divided into two, there is within the crystal a fixed direction , 
not depending upon the shape of the surface , so that the directions of both refracted rays 
