( 762 ) 
Mathematics. — “On a system of conics in space.” By Mr. Lucien 
Gopraux of Liege. (Communicated by Prof. P. H. Scnourw). 
In this note [ study a five times infinite system formed by conics 
in space, related to six connexes (point-plane) of the first order. This 
system of conics is in birational correspondence with the system of 
elements of space, each of which consists in a-line and a plane 
passing through it. The conic corresponding to the combination of a 
line / and a plane + passing through it lies in the plane a, and a 
definite quadratic transformation of this plane into itself transforms 
the conic into the line / It is in the definition of this quadratic 
correspondence that the given connexes enter. 
1. Let be given to us in space two triplets of mutually independent 
connexes of the first order: ®,, ®,, ®,, Y\, U, W,. The equations of 
these connexes are respectively : 
g,(@,u=2,9, (+ - - - - +49, H=9, 
Pp, (, u) ==, Pp, (U) + .. dr (6) == 0; 
CG At, HIE plays cass Ee Oe 
ab, (a, u) 2, W,U) Hon en TE Wi =D 
ab. (ce, uh =D (uy) on oc oe. Heh OS 
| 
w, (a, u) =a, Wo, (U) + 
f 
point-coordinates being represented by (w,, 2, .7,a,) and tangential 
coordinates by (2,, u, Ws, U,). 
Let m be the class of the first triplet of connexes and n that of 
tbe second, the functions gi (u) being of order m in the coordinates 
(u, Us, Ua, Us) and the functions Wz (wz) of order 7. 
Let us consider the general plane 
Us =U, rt + Ut, + U, 2, pu, =O0. sn oe 
The points which combined to this plane satisfy the equation of the 
connex ,, form another plane meeting (u) in adine ¢,. Likewise 
the connexes ®,, ®,, U, U, U, determine in the plane (7) the lines 
i ene 
Let us imagine a quadratic transformation of the plane (u) in 
itself, transforming any line of that plane into a conic circumscribed 
to the triangle formed by the lines @,, @,,@,. In order to define this 
transformation completely we suppose that to the lines 9,8, 8, 
correspond respectively the degenerated conics formed by the lines «, 
B, 
and «,, «‚, and @,, «, and a, 
