56 Prof. Plilcker on a New Geometry of Space. [Feb. 2, 



plane. There is, with regard to each complex passing through the congru- 

 ency, one right line conjugated to a given one. All these conjugated lines 

 constitute one generation of a hyperboloid, while the right lines of its 

 other generation are rays of the congruency, which therefore may be gene- 

 rated by a variable hyperboloid turning round one of its right lines. 



The axes of all complexes intersecting each other along a linear con- 

 gruency meet at right angles a fixed line, which is the axis of the con- 

 gruency. Among the complexes there are especially two, the axes of 

 which are met by their rays. These axes, meeting themselves the axis of 

 the congruency, are its directrices. A linear congruency, depending upon 

 eight constants, is fully determined by means of its two directrices. Each 

 right line intersecting both directrices is one of its rays. The plane par- 

 allel to both directrices, and at equal distance from them, is the central 

 plane of the congruency; the point where it meets r under right angles, the 

 axis of the congruency, its centre. The two lines bisecting within the 

 central plane the projections of the two directrices, are its secondary axe. 

 The directrices may be as well both real as both imaginary. In peculiar cases 

 the two directrices are congruent, or one of them is at an infinite distance. 

 Each of two complexes being given by means of its axis and its constant k f 

 both directrices of the congruency along which they intersect one another 

 are analytically determined. A congruency being given by means of its 

 directrices, the constants and axes of all complexes passing through it are 

 determined. 



A linear configuration of rays is the common intersection of any three 

 linear complexes, and represented by their equations, 11=0, ii'=0, Q"=0. 

 Each complex represented by an equation of the form ! + /*}' + !' =0, 

 equally passes through the same configuration. So does any congruency 

 along which two such complexes meet. A linear configuration is a hyper- 

 boloid ; its rays constitute one of its generations, while the directrices of 

 all traversing congruencies constitute the other. The central planes of all 

 these congruencies meet in the same point the centre of the hyperboloid. 

 Its diameters meet both directrices of the different congruencies. The 

 directrices are either real or imaginary ; accordingly the diameters meet 

 the hyperboloid, or meet it not. If the two directrices are congruent, the 

 diameters become asymptotes. The hyperboloid passes into a paraboloid 

 if there is one directrix infinitely distant. 



A linear configuration is determined by means of three congruencies as 

 it is by means of three complexes. That ray of it which meets one direc- 

 trix of each congruency is parallel to the other. By drawing two planes 

 through the two directrices of each of the three congruencies parallel to 

 its central plane, we get a rhomboid circumscribed about the hyperboloid, 

 the points of contact, within the six planes, being the points where the six 

 directrices are intersected by the rays. A hyperboloid being given, we 

 may revert to the congruencies and complexes constituting it. Finally, 

 the equation of the hyperboloid in ordinary coordinates, oc, y, s, is derived. 



