which equations determine d 1 and 0 2 . 
Writing the equations under the form 
and treating d 1 and 0. 2 as coordinates, each of these equations belongs to a curve of 
the order 2 (to — 1), having a (to — l)thic point at infinity on each of the axes. The 
number of intersections thus is 
= 4 (to — l) 2 — (to — l) 2 — (to — l) 2 , = 2 (to — l) 2 . 
But among these are included points not belonging to the original system, viz. the 
points for which (A x = 0, A 2 = 0) other than those for which 0 X = 6. 2 ; the points so 
included are in number = to 2 — to ; and omitting them, the number is 
(2 (to — l) 2 — to (to — 1)) = [to — l] 2 , 
which is the number of points 0 X lying in lined with the origin and another point 
6. 2 ; the number of apparent double points is the half of this, or h = \ [to — l] 2 . And 
thence 
M = (— i [to] 2 + h =) — (to — 1). 
I investigate also the number of lines through two points which meet two 
arbitrary lines ; this is in fact = S (1, to 2 ), which for the curve in question is 
= (i M 2 - (to -1) =) (to - l) 2 . 
Let the equations of the two lines be (x = 0, y = 0) and (z = 0, w = 0) ; then the con 
ditions to be satisfied are 
A 1 = B 1 C 2 _ A. 
A 2 jB 2 ’ Go DA 
or writing these under the form 
a x b 2 - a 2 b 2 
and treating 0 1} 6 2 as coordinates, the number of intersections of these two curves is 
= 2 (to— l) 2 , the same as for the two curves last above considered. And the number 
of the lines in question is one half of this, or = (to — l) 2 . 
f the to roots, 
ìe symmetrical 
3 the required 
(Aj, B 2 , C 1} D x ) 
ts, the form is 
— di), or II is 
i M 3 =) h 
s of the roots