788 
DR. PLUCKER ON A NEW GEOMETRY OF SPACE. 
cularizing step by step from the general complex to a surface of the second order and 
class, determined by its tangents. 
I intend resuming the consideration of the mechanical part of this note. Then a last 
generalization will occur to us, the equation of condition, hitherto admitted between 
the six coordinates x, y, z, L, M, N, being removed. 
CONTENTS. 
I. On Linear Complexes of Might Lines. 
Preliminary explanations. — Point-coordinates. Equations between them representing 
surfaces by means of their points. Plane coordinates. Equations between them repre- 
senting surfaces enveloped by planes, 1. Double definition of right lines, either by 
means of their points or by means of traversing planes. Pays. Axes. The two pro- 
jections of a ray within two planes of coordinates depend upon four linear constants, 
which may be regarded as ray-coordinates, r, s, g, <r and t, u, v x , v y . The two points in 
which two planes of coordinates are intersected by an axis, depend upon four linear 
constants which are its coordinates, x, y, z t , z u and p, q, vr, z, 2-5. Complexes of rays or 
axes represented by one equation between their four coordinates. Congruent lines of 
two complexes constitute a congruency, of three complexes a configuration ( surface 
gauche). In a complex every point is the vertex of a cone, every plane contains an 
enveloped cone. In a congruency there is a certain number of right lines passing 
through a given point, and confined within a given plane, 6, 7. A configuration of rays 
represented by three linear equations, either between r, s, g, a or £, u, v x , v y , is a para- 
boloid, or a hyperboloid, 8. A configuration of axes represented by three equations, 
either between p, q, -a. z or x , y , z t , z u , is either a hyperboloid or a paraboloid, 9. In a 
congruency of rays or axes represented by two linear equations, there is one ray and one 
axis passing through a given point and confined within a given plane, 10. Construction, 
by means of two fixed points, of the rays of a congruency represented by two linear 
equations between t , u, v x , v p 11. Construction, by means of two planes, of the axes of 
a congruency represented by two linear equations between x, y, z t , z u , 12. 
Linear complexes of rays. — In a complex represented by a linear equation between 
r, s, g, <r, all rays traversing a given point constitute a plane ; all rays confined within a 
given plane meet in the same point. Points and planes corresponding to each other 
13-15. A new variable (sg— re) introduced. The general equation of a linear complex 
is Ar+Bs-f-C+D<7-|-E^+F(^— r<r)=0. Equation of a plane corresponding to a given 
point, of a point corresponding to a given plane, 16-19. Conjugate right lines. 
Each ray intersecting any two conjugate lines is a ray of the complex. A ray of the 
