790 
DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
An infinite number of congruencies meet along a linear configuration. Generally it is a 
hyperboloid. Its rays constitute one of its generations, while the directrices of all con- 
gruencies constitute the other, 62. The central planes of all congruencies meet in the 
same point : the centre of the configuration. Its diameters meet both directrices of the 
different congruencies, 63. A configuration is determined by means of three complexes, 
or by means of three congruencies, obtained by combining them two by two. Three 
couples of planes drawn through both directrices of each of the congruencies parallel to 
its central plane constitute a parallelopiped circumscribed to the hyperboloid. Each ray 
intersects all six directrices. The ray within each of the six planes is parallel to the 
directrix within the opposite plane ; the point in which it meets the directrix within 
the same plane is the point of contact. Three diameters determined by both points of 
contact within the three couples of opposite planes. Imaginary diameters correspond 
to imaginary directrices, asymptotes to congruent directrices, 64. A hyperboloid being 
given, we may return to the congruencies and complexes which constitute it, 65. The 
equations of the configurations transformed into an equation between x , y, z, 66. 
II. On Complexes of Luminous Bays within Biaxal Crystals. 
Complexes of doubly refracted rays corresponding to complexes of incident rays, 1. 
Digression on double refraction. Huyghexs’s principle. The author’s construction 
presented, 1838. Auxiliary ellipsoids. The ellipsoid E, with regard to which the wave- 
surface is its own polar surface. The plane of refraction, containing both refracted 
rays, passes through SS, the polar line of RR, along which the surface of the crystal is 
intersected by the front of the incident elementary wave at that moment when, within 
the crystal, the wave-surface is formed, 2-6. Tlie plane of refraction is congruent with 
the diametral plane of E, the conjugate diameter of which is perpendicular to the 
plane of incidence hi O, 7. All rays incident within the same plane are, after double 
refraction, confined again within the same plane, 8. While the plane of incidence 
turns round the vertical in O, the corresponding plane of refraction turns round that 
diameter of E, the conjugate diametral plane of which is the surface of the crystal, 9, 
10. Whatever may be the plane or curved surface met by an incident ray in any given 
point O, all corresponding planes of refraction pass through a fixed right line, 11. 
Peculiar cases of complexes. The plane of refraction perpendicular to the surface of 
the crystal. The incident and the two refracted rays confined within the same plane. 
A circular section of E falling within the surface of the crystal, 12, 13. Analytical 
determination of SS, 14. A fourth auxiliary ellipsoid, 15, 16. 
Complex of doubly refracted rays determined by means of E. Its equation depend- 
ing upon the constants of E, 17, 18. By taking as axes of coordinates three conjugate 
diameters of E, two of which, falling within the surface of the crystal, are perpendicular 
to each other, the general equation of the complex becomes ra—hs^ the constant Jc 
being the ratio of the squares of the two rectangular diameters, 19. Geometrical in- 
terpretation, 20. Refracted rays of the complex passing through a given point consti- 
