( 764 ) 
line in that plane corresponds a conic in that plane, and reversely, 
for we can pass from the equations (¢) to the equations: 
JP US gees 0) 
So the conics ¢ form a five times infinite system i. 
De Toa Aine J 
i ==): Or =F TAG ve telen OER 
corresponds obviously a one time infinite system of conics of /’; we 
now try to find the order of the surface S forming the locus of this 
system of conics. 
Let us put 
ja; +4b; 5 (ERE 
and apply this substitution to the first of the equations (¢). By sub- 
As. 4,, the first row from the 
second, these determinants become of order 2 in 4; so we obtain 
an equation of order 2(m-+- 7) in 4 By substituting in this equation 
for à the value derived from 
dat £0, 0; 
we find the equation of the locus JS. 
tracting in the determinants A, A,,, 
Remarking tbat any plane through / meets S in one conic only, 
we find the following theorem : 
The conics corresponding to any line generate a surface Sof order 
2m + n+ 1) passing 2m +n) times through that line. 
3. Let us now determine the locus of the lines in space, 
corresponding to conics of / passing through a given point P. 
Obviously this locus is a complex, which may be represented by J7. 
As we started from connexes ®, ¥ the equations of which are quite 
general, it will be sufficient, in order to determine the order of J/, 
to find the numoder of lines in the plane wx 
; — O passing through the 
point „2, = 2, = 2,0, to which correspond comes of U passim 
through the point Pv, = «, = 2, = 0). This introduces the relations 
ul, = U, = 0, a,=0 into therequations (¢ of art. 1. Moreover me 
may suppose without any restriction that @, disappears too, for in 
the equations (e) the parameters a a, reduce themselves to 
three homogeneous parameters. So the condition that the planes 
Us Le + Uy din Orde Lit Ost, == 
intersect according a line of the plane x, =O is 
ities ne 
eyo SOE eo eae 
