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



we shall distinguish by the name of ray, may be determined by means of 

 two projections, for instance by those within XZ and YZ, represented by 

 x = rz + P> 



or by 



In admitting the first system of equations, a ray is determined in a linear 

 way by means of the four constants r, s, p, a, which may be called its four 

 coordinates, two of them, r and *, indicating its direction, the remaining 

 two, after its direction being determined, its position in space. In adopting 

 the second pair of equations, t, u, v xt v y will be the coordinates of the ray. 



A right line of the second description, which we shall distinguish by the 

 name of axis, is determined by any two of its points. It is the common 

 intersection of all planes passing through both points. We may select the 

 intersection of the axis with the two planes, XZ and YZ, as two such 

 points, and represent them by 



net +z t v= I, 



yu+z u v=l, 

 or by 



In making use of the first pair of equations, the four constants x, y, z t) z u , 

 indicating the position of the two points within XZ and YZ, are the coordi- 

 nates of the axis. In adopting the second pair, the four coordinates of the 

 axis are p, q, IT, K. 



A complex of rays or axes is represented by means of a single equation 

 between their four coordinates ; a congruency, containing all congruent 

 lines of two complexes, by means of two such equations ; a configuration, 

 containing the right lines common to three complexes, by three equations. 

 In a complex every point is the vertex of a cone, every plane contains an 

 enveloped curve. In a congruency there is a certain number of right 

 lines passing as well through a given point as confined within a given 

 plane. A configuration is generated by a moving right line. 



In a linear complex the right lines passing through a given point con- 

 stitute a plane ; all right lines within a given plane pass through a fixed 

 point. Two linear complexes intersect each other along a linear con- 

 gruency. In such a linear congruency there is a single right line passing 

 as well through a given point as confined within a given plane. Three 

 linear complexes meet along a linear confguration. 



Instances of linear complexes are obtained by means of linear equations 

 between the four coordinates of any one of the four systems. A linear 

 configuration of rays represented by three such equations between r, s, p, a 

 is a paraboloid, immediately obtained ; between t, u, v x , v y a hyperboloid. 



