646 
the points of contact of the 6 tangents drawn out of O to y,, and 
the + points of intersection of z with y,. 
If we now put 
r=—-—], 
we choose out of the pencil a curve which has degenerated into 
the two lines w and y and a conic } with the equation 
B= x* may 4+ y’?— 2? =0. 
8 is called “the conic of Brrtint’. 
On the cone B are situated the points of contact of the 6 tangents 
drawn out of O to y,, and the two points of intersection of y, 
with z. 
If we further draw through O an arbitrary line 
y—l«=0, 
its points of intersection with y, are found from 
Use? (1 + ml + 1?) + 27} + we (a 4 bl 4 cl? + dl’) — 0, 
those with @ from 
a? (1 4+ ml + [)— 27? =0, 
from which appears at once: 
Any line | through O intersects y, besides in O in 2 more points 
which are harmonically separated by the points of intersection of l 
with B. 
2. The curve in question is now obtained by putting in (I) the 
coefficient m equal to zero. The geometrical significance of this is 
the following : 
The curve y, considered is cut by the line z joining the two other 
points in which y, ts intersected by its tangents at the node, in two 
more points, harmonically separated by the former two. 
For convenience’ sake I shall speak in the future of the “harmo- 
nical uninodal curve y,”. 
Its equation is: 
Y, = wy (a? + y? + 27) + 2 (ar? + bx*y + cry? 4+ dy*)=0. (1) 
If we now put 
1=@ 1+¢ 
g=(1+C)a* + (1—C)y? + 22 |«( ih me) tele nd 
and 
en 4a i ime Ad 
EO 
in which C is an arbitrary constant, we get: 
re OER “CO. 2 5. 
