58 Prof. Pliicker on a New Geometry of Space. [Feb. 2, 



1838 I promised a discussion of the construction given, but neglected it 

 till the present time. This discussion immediately leads us to represent 

 the complex of double-refracted rays by an equation, and at the same 

 time we meet with several theorems worthy of notice. 



If there is any incident ray, the plane of refraction, containing both 

 double-refracted rays, is congruent with the diametral plane of the aux- 

 iliary ellipsoid, the conjugated diameter of which is perpendicular to the 

 plane of incidence. All rays incident within the same plane are, after 

 double refraction, confined again within the same plane. While the plane 

 of incidence turns round the vertical, the corresponding plane of refraction 

 turns round that diameter of the auxiliary ellipsoid, the conjugated dia- 

 metral plane of which is the surface of the crystal. Whatever may be the 

 plane or curved surface by which a crystal is bounded in a given point, 

 all corresponding planes of refraction pass through a fixed right line. 



A complex of rays starting in air in all directions from every point of a 

 luminous right line, perpendicular to the surface of the crystal, is repre- 

 sented by the equation 



ra=sp, 



the luminous right line being the axis OZ, while the two remaining axes, 

 OX and OY, are within the surface of the crystal any two right lines per- 

 pendicular to each other. This complex is transformed by double refrac- 

 tion into another, the equation of which assumes the most simple form, 



rafcsp, 



in especially admitting that the two axes, OX and OY, are congruent with 

 the axes of the ellipse along which the auxiliary ellipsoid is cut by the surface 

 of the crystal, and that the third axis, OZ, is, within the crystal, the dia- 

 meter of the axixiliary ellipsoid ; the conjugated diametral plane is that 

 surface. A is a constant indicating the ratio of the squares of the two 

 axes of the ellipse. 



The complex of double-refracted rays is of the second order ; its equa- 

 tion may be easily submitted to analytical discussion. All its rays pass- 

 ing through any given point constitute a cone of the second order. This 

 cone remains the same if the point describes a right line, passing through 

 the origin. Likewise there is in any given plane a hyperbola, enveloped 

 by rays of the complex. Peculiar cases are easily determined. The com- 

 plex of double-refracted rays may be described in three different ways by a 

 variable linear congruency. In the peculiar case in which the surface of 

 the crystal is a principal section, OZ becomes perpendicular to it ; if it is 

 one of the circular sections of the auxiliary ellipsoid, the constant k be- 

 comes equal to unity, i. e. all double-refracted rays meet the axis OZ. 

 From the general case the case of uniaxal crystals is immediately derived. 



