647 
For this reason we can produce y, as the locus of the intersections 
of corresponding curves of 2 projective pencils of conics: 
Pa dw=0 Ne eten te (9) 
and (!1—C) a? + (1+ C)y?— 2aay = 0, . . . . (A) 
where the pencil (4) consists of an involution of rays with O as 
centre. 
3. We will now first give a geometrical interpretation of the 
way in which the projectivity between (3) and (4) has been fixed. 
By putting 4=/, we choose an arbitrary conic from (3) inter- 
secting z in the points A and A’ defined by: 
(1+ C) aw + 2a,ay + (I—C) y? = 0. 
We put moreover 
(LC) x* + 2a,ey + (L—C) yr =(1—C) (ype) Ype), 
so that : 
1+ C=(1—C) pp’ and 2/4, =(1—C)(p + p’). 
The points B and B’, harmonically separated from A and A’ by 
the points of intersection of z and 8 are then respectively found 
from : 
ce Di) 
and « — p'y= 0 
Hence the pair of rays which project the points B and B’ out 
of O, have for equation: 
C= == 
or 
(LC) 2? + (14 C) y? — 2),cy = 0. 
For this reason the projectivity between (B) and (4) is fived in this 
way: a pair of corresponding conics of (3) and (4) always intersect 
z in pairs of points (A, A’) and (B, B’) so that A and B and also 
A’ and B’ are harmonically separated by B. 
4. Concerning the pencil (3) we remark that both p and yw are 
harmonically circumseribed to p, i.e. circumscribed to an infinity of 
polar triangles of B, as appears directly from this that one of the 
simultaneous invariants, — generally called @ —, formed out of the 
coefficients of 3 and p (or ws), becomes zero. 
From this follows: 
1. each conic of (3) is harmonically circumscribed to B; 
2. the base points of the pencil (3) form a polar quadrangle of B, 
ie. a quadrangle of which each side passes through the pole of the 
opposite side with regard to B. 
