1865.] Prof. Plucker on a New Geometry of Space. 53 



February 2, 1865. 



Major-General SABINE, President, in the Chair. 

 The following communications were read : 



I. " On a New Geometry of Space." By JULIUS PLUCKER, For. 

 Memb. E.S. Received December 22, 1864. 



(Abstract.) 



Infinite space may be considered either as consisting of points or trans- 

 versed by planes. The points, in the first conception, are determined by 

 their coordinates, by x, y, z for instance, taken in the ordinary significa- 

 tion ; the planes, in the second conception, are determined in an analogous 

 way by their coordinates, introduced by myself into analytical geometry, 

 by t> u > v f r instance. The equation 



represents, in regarding x, y, z as variable, t, u, v as constant, a plane by 

 means of its points. The three constants t, u, v are the coordinates of 

 this plane. The same equation, in regarding t, u, v as variable, x, y, z as 

 constant, represents a point by means of planes passing through it. The 

 three constants x, y, z are the coordinates of this point. 



The geometrical constitution of space, referred hitherto either to points or 

 to planes, may as well be referred to right lines. According to the double 

 definition of such lines, there occurs to us a double construction of space. 

 In the first construction we imagine infinite space to be traversed by lines, 

 themselves consisting of points ; an infinite number of such lines in all 

 directions pass through any given point ; the point may describe each of 

 the lines. This constitution of space is admitted when, in optics, we 

 regard luminous points sending out in all directions rays of light, or, in 

 mechanics, forces acting on points in any direction. In the second con- 

 struction, infinite space is regarded likewise as traversed by right lines, but 

 these lines are determined by planes passing through them. Every plane 

 contains an infinite number of lines having within it every position and 

 direction, round each of which the plane may turn. We refer to this 

 second construction when, in optics, we regard, instead of rays, the corre- 

 sponding fronts of waves and their consecutive intersections, or when, in 

 mechanics, according to Poinsot's ingenious philosophical views, we intro- 

 duce into its fundamental principles " couples," as well entitled to occupy 

 their place as ordinary forces. The instantaneous axes of rotation are 

 right lines of the second description. 



The position of a right line depends upon four constants, which may be 

 determined in a different way. I adopted for this purpose the ordinary 

 system of three axes of coordinates. A line of the first description, which 



VOL. XIV. F 



