495 
now the point of intersection P of 6, and a is to coincide with the 
point of intersection (} of 6, and a then the four points of £* in the 
plane «b,b, must form a complete quadrangle with P and 7, as two 
of the three diagonal points, and this is only possible if the line 7’, P, 
thus a, lies on a special cone of order two, which will in general 
not be the case. In an arbitrary plane through 7, lie namely four 
points of £*, forming a complete quadrangle ; one of the three dia- 
gonal points is 7’, the two other ones lie in 7,7,7’, and evidently 
describe here when the plane varies a conic through 7), 7,7’. If now 
a happened to lie on the cone projecting this conic out of 7), then 
two coincidences of the (2,2) correspondence would lie on the conic 
and the two others in 7,; in every other case however all four 
coincidences must coincide in 7’, and so « must touch the surface 
OF ined: 
4. We now proceed to determine the points of intersection of 
2° with an entirely arbitrary line /. To that end we allow a point 
P to travel along the line / and we investigate how often the chord 
a passing through P lies in plane Ps. According to $ 3 the chords a 
issuing from the points 2? of / form a scroll of order four with 
nodal curve 4* and single directrix /; the lines s belonging to the 
points P of / form a regulus and the planes Ps envelope a develop- 
able of class 3. If namely point P describes the line / then the two 
polar planes 2, and a, of P with respect to ®, and ®, (comp. § 2) 
revolve around the two lines /,,/, conjugated to /, and crossing each 
other in general; thus the line s describes a regulus with l, and l, as 
bearers. 
Now the surface enveloped by the planes Ps. We imagine an 
arbitrary point V in space, we choose a point P on /, we determine 
the corresponding line s and we find the point of intersection Q of 
the plane Os with /; in this manner to each point P one point Q 
corresponds. If reversely we wish to know how many points P cor- 
respond to @Q, we draw the line connecting O and Q and we 
intersect it by the regulus of the lines s just found; through each 
of the two points of intersection passes one line s whose corresponding 
point P lies on /, so that to one point Q two points P correspond. 
Between the points P and Q on / there exists a (1, 2) correspond- 
ence; for the three coincidences the plane Ps passes through O; 
so the planes Ps belonging to the points of a line L envelope a 
developable of class three. 
We now add to the figure an arbitrary plane « and we determine 
the section of this plane with the scroll of order four, formed by 
