780 
DE. PLUCKEE ON A NEW OEOMETEY OE SPACE. 
point in space and therefore regarded as constant, while x, y , z are the variable coordi- 
nates of the points of any line of the complex, that equation represents a cone of the nth. 
order, the geometrical locus of lines of the complex passing through the given point. 
Hence 
A complex of the nth degree being given , each point of space is the centre of a cone of 
the nth order into which lines of the complex converge. 
In linear complexes the lines meeting in a given point constitute a plane. If n— 2, 
the cones are of the second order, and may degenerate into two real or imaginary 
planes. 
14. The right lines constituting a complex may be distributed either within planes 
traversing space, or according to points into which they converge. We hitherto con- 
sidered as a complex of right lines, the number of which is oo 3 . We may as well 
regard it either as a complex of curves, or as a complex of cones, the number both of 
curves and cones being oo 2 . Therefore we may say that 
O„=0 
represents at the same time as well in each plane a curve of the nth class as cones of the 
nth order having each point of space as centre. 
The curve in a plane revolving round a given line, or moving parallel to itself, gene- 
rates a surface. The cone the centre of which describes a given right line envelopes 
a surface. The number of surfaces both generated by the curve and enveloped by cones 
is co. There is one of each kind of surfaces corresponding to any given line, all sur- 
faces will be exhausted if that line turns in all directions round any of its points. 
Accordingly we may likewise consider as a complex of surfaces, either described by 
curves or enveloped by cones. 
15. In denoting by g> any constant coefficient, 
O„+^O m =0 (3) 
represents an infinite number of complexes. The lines congruent in any two of them 
belong simultaneously to all. All these congruent lines constitute a congruency (Q„, Q m ), 
which we say is represented by the equations of the two complexes. 
• Each plane traversing space confines a curve of each of the two complexes, the mn 
tangents common to both curves belong to the congruency. All curves within the same 
plane belonging to the different complexes (3) which pass through the congruency, 
touch the same mn of its lines. Again, each point is the centre of a cone belonging to 
the different complexes (3). All such cones meet along the same mn[ lines, likewise 
belonging to the congruency. Therefore in a congruency (Q„, Q m ) there are mn lines 
confined within each plane as there are mn lines passing through each point. The num- 
ber of lines constituting a congruency is oo 2 . 
If m— 1, there are in each plane n lines of the congruency (£2„, OJ passing through 
the same point, as n of its lines converging into each point fall within the same plane ; 
plane and point corresponding to each other. 
