( 763 ) 
Let us try to form the equation of the conic & corresponding to 
the line represented by the equations 
neem VE Oe rt NA 
and let us apply to the whole space the birational transformation 
(T), defined by 
oy ety UY, Uit Uy, yu Hug pug). {T) 
Then the plane w, =O becomes the plane vy, = 0, and the equation 
of the line @', corresponding to @, runs 
1 7 3 ae feet 
Le =D Ui Pr) Ys (Us Pi UP) FY s(t Pia Ui Pr =O. (ee). 
The lines «,, a,, B,, B, B, ave transformed into the lines @’,,a’,,.8', 808. 
characterized by analogous equations. 
The equation of the line / corresponding to / is 
y, (uw, a, —u,a,) + y, (Uy a, —u, A) + 43 (U, A3 — ua.) = 9. (U) 
So the quadratic transformation of the plane (u) in itself becomes 
an analogous transformation of the plane y,—=O in itself, any line 
OON am LP met) i a se es ey (5) 
being transformed into a conic with the equation 
1 5 a Pr Í s . a eA ey [= 
(BO HRabe Anbo) Ile sd ~ - - - (4519.55.09. + 45305) Le, Ile. |=9.(1). 
The undetermined parameters £ may be eliminated by means of 
the remark, that if the line (4) coincides successively with 8, 8, 3',, 
the corresponding conic (1) degenerates into the pairs of lines @’, and «',, 
and «',. So we easily find for the equation of the 
conic (&') corresponding to the line /: 
= = fi ! ! (ny (pa € 
aye Le] [e's ] aie A. [es] KA EE pave | ce | [es] — 0, - (2) 
if A,,,A,,,4,, are defined by 
! | ! 
Gee BN) ee ate 
ipa 1 2 3 4 p. 
An im / Wa, Wes Wy. Wo, an ai a a 
Wi. Wan Wes Wee | 
Obviously the conie (€) corresponds to the conic (d) obtained by 
transforming / in the plane (wv); so the equations of this conic will 
be found by applying to (2) the transformation (7!) represented by: 
MeL. Oy LE Deen IL) 
So after some reductions we find: 
An wp, UW; SIT AGN Yu. Ww, - re w, wy, zE 0, | 
) 
ier ss) \ (e 
which equations show that to the combination of a plane and a 
