499 
the chords of /* resting on /, as well as with the developable just 
found of class three; the former is a rational curve of order four 
with three nodes in the points of intersection of « and /* and a 
single point in the point of intersection of @ and /, the second a 
rational curve of class 38 with a double tangent. 
Through an arbitrary point of the curve of order four passes one 
chord a, intersecting / in P, and through P passes one plane Ps, 
so that in this way to each point of the curve /* of order four one 
tangent of the curve &, of class three corresponds, whilst in the 
same way we can see that to a node of /* two different tangents 
of &, correspond. In the same easy way we can convince our- 
selves that to each tangent of 4, one point of &* corresponds and 
to the double tangent two different ones; so the result is that there 
exists a (1, 1) correspondence between the points of 4‘ and the tangents 
of £,; the question now is how many coincidences this correspondence 
possesses. . 
Let us take a point P on £* and let us determine the corresponding 
tangent ¢ of &,, cutting /* in four points Q; reversely through one 
point Q pass three tangents ¢, and to each of these one point P 
corresponds; so between the points P and Q exists a (3, 4) corre- 
spondence and, as the bearer is rational, the number of coincidences 
is seven. One of these must necessarily be the point of intersection 
of land «; for, through this point taken as point P of /, passes a 
chord a and likewise a plane Ps cutting « of course according toa 
line passing through P, however without it being necessary for a 
to lie in the plane Ps. So we have here a coincidence in the plane 
« to which no incidence of a into the plane Ps corresponds; if we 
set this case apart six coincidences remain which are each the conse- 
quence of a point of intersection of / and 2°. 
For the sake of completeness we add to the preceding that the 
regulus of the rays s belonging to the points P of / contains the 
four vertices of the’ cones 7,,..., 7, (comp. $ 3); for 7,’ has’ as 
polar plane witb respect to ®, as well as to ®, the plane 7',7',7,, 
so inversely the two polar planes of the point of intersection of / with 
this plane pass through 7’,, and so does therefore their line of 
intersection s. 
The developable of the planes Ps is of class three, «so through 
each point P of / itself three planes Ps must pass; indeed two 
rays s of the regulus cut / and to these two points P of / cor- 
respond; so through / pass two planes Ps and these must for 
each point of 7 be added to the plane passing through that point 
but not through /. 
33* 
