( 656 ) 
Retaining the three lines 
20 
ae, bt, == 0 
and 
4 mna, — (a'a, — bw) = 0, 
we can change the fourth rav, the equation of which is of the form 
(10). The eross ratio of these four lines will be: 
EN 
therefore the value of & is independent of the curve k,, and is a 
function of k alone. 
Eh 
4, We have seen, that in the projective generation of /, to any 
ray of pencil |/| correspond two conics of the involutory pencil. 
The values of the parameters u for these conics, which correspond to 
the right lines, indicated by (10), will be determined out of the 
equation, with respect to (3): 
(2k)?—1 mi u 
Amn (n— my): 
This quadratic equation furnishes two pairs of values for u, namely 
n [2k 2 1} 
m [2k == 1] 
We can now determine any number of discrete points of 4, as 
follows; putting 
(41,2 in 
p=a'a, — ba, — 0 
we can write 
4 mna, + [2k + 1] [2k —1] p=—0 
| 0 Sh od 
m 2k) se. Me Enk 1m) 
m,n being whatever constant numbers and p any right line passing 
through A‘). If we eliminate x, out of the equations of the latter 
system, we shall obtain two pairs of pencils with non-consecutive rays 
in a correspondence (1, 2) i.e. [A], [A], having the equations 
A mna, + [24 + 1] [24 — 1] p= 
Ame, — [2h se A? pe, = 90 
where the coefficients have an interesting form. 
| = 0 Ieee): 
1) In my paper: Ein Satz über die ebene Kurve 4. Ordnung mit einer Spitze 
2. Art, Sitzungsberichte der K. Akademie in Wien, Ila, CXIX, 1910, I have con- 
sidered a few similar relations for this curve of deficiency one. Next time I 
shall treat the same relations for a quartic curve with a spinode and a rhamphoid 
cusp (deficiency zero). 
