( 766 ) 
The locus of the lines to which correspond conics of E passing 
through a given point P and lying in planes passing through a 
given line p through P is a congruence a of order unity and. of 
class 4(m+n+1)+14, having the line p and a curve of order 
4(mtn+1)+1 for singular lines. 
We still remark that any two lines of the complex 47, which do 
not belong to a same pencil lying in a plane through P, determine 
one and only one congruence a contained in JZ. So: 
The complex MH contains a net of congruences x. 
5. It may happen that the line indicated by «a, in art. L is not 
univocally determined for some particular positions of plane 1); 
this will be the ease for the planes (7) satisfying the equations 
[| gir (u) pia (Ww) gis (w) pis (4) 
sel |e (5) 
u, U, Us u, 
For under these conditions the plane defined by the connex ®, 
coincides with the plane (u) and we can take for the line «‚ any 
line of this plane. In this case the quadratic transformation of the 
plane (wv) in itself presents two degrees of indetermination; so we 
find: Po any line of a plane (uw) given by the equations (5) correspond 
oo? conics of the plane. 
The equations (5) are satisfied by m* + m? + m + 1 planes. 
Repeating the same reasoning for the connexes ®,, ®, we find: 
There are 3 Gn + D (m? + 1) planes with the property that to each 
line of the plane corresponds a double infinity of conics of the plane. 
Any plane of space generally contains a net of conies of /; if 
this plane satisfies the equations 
Wy (xe) io (U) W.3 (1) ia (w) 
u u u u 
1 2 8 4 
it is clear that any conic contained in it may be considered to 
correspond to an arbitrarily chosen line of the plane. So: 
There are B mn +1) (n? + 1) planes with the property that to each 
line of the plane corresponds a net of conics m the plane, this net 
being the same for all the lines of the plane. 
Any plane satisfying either the equations (5) or the analogous 
ones for ,, ®,, or the equations (6) is obviously principal for any 
complex 4 with respect to any point P of that plane. 
Morlanwelz, December 1910. 
