1865.] Prof. PI ticker on a New Geometry of Space. 55 



A linear configuration of axes represented by three linear equations between 

 p, q, IT, K is a hyperboloid, immediately obtained ; between x, y, z t , g u a 

 paraboloid. Instances of linear congruences are exhibited by means of 

 two linear equations, as well between t, u, v x , v y as between x, y, zt, Z M and 

 their right lines easily constructed. 



The general linear equation, however, between any four coordinates does 

 not represent a linear complex of the most general description. Besides, 

 there is a want of symmetry, the four coordinates depending upon the 

 choice of both planes, XZ and YZ. This double inconvenience, if not elimi- 

 nated, would render it impossible to adapt in a proper way analysis to the 

 new geometrical conception of space. But it may be eliminated in the 

 most satisfactory way. 



For that purpose I introduced (in confining myself to the case of the co- 

 ordinates r, s, p, a) a fifth coordinate (spra), which is a function of the four 

 primitive ones. Then the linear equation between the five coordinates 



is the most general of a linear complex. After having been rendered 

 homogeneous by a sixth variable introduced, it becomes of a complete 

 symmetry with regard to the three axes OX, OY, OZ. The introduction of 

 the fifth coordinate (sprv) is the real basis of the new analytical geo- 

 metry, the exploration of which is indicated in the ordinary way. 



In the paper presented, a complete analytical discussion of a linear 

 complex is given. We may for any point of space construct the corre- 

 sponding plane containing all traversing rays, and vice versa. Right lines 

 of space associate themselves into couples of conjugated lines ; to each 

 line a conjugated one corresponds. Any right line intersecting any two 

 conjugated, is a ray of the complex. Each ray of it is to be regarded as 

 two coincident conjugated lines. It is easily shown that each linear com- 

 plex may be represented by means of any one of the following three 

 equations, in which Jc indicates the same constant : 

 sp ra = k, <r=kr, p=ks. 



Accordingly a linear complex depends upon the position of a fixed line 

 (depending itself upon four constants) and the constant k. Hence it like- 

 wise follows that such a complex of rays may, without being changed, as 

 well turn round that fixed line, the axis of the complex, as move along it, 

 parallel to itself. The same results may be confirmed by means of the 

 transformation of ray-coordinates, and thus analytically determined by the 

 primitive constants A, B, C, D, E, F, the position of the axis of the com- 

 plex and its constant k. In a peculiar case, where k becomes zero, all 

 rays of the complex meet its axis. 



A linear congruency of rays, along which an infinite number of linear 

 complexes meet, is represented by the equations of any two of these com- 

 plexes. Through a given point of space passes only one ray, correspond- 

 ing to it, as there is only one corresponding ray confined within a given 



F2 



