505 
which ensues that the twenty common chords of hè and k* are single 
lines of &,,. 
The plane a issuing from a point of %* contains two chords a 
and so it counts twice as tangential plane of 2,,, whilst reversely 
it is easy to see that @,, can have no other double tangential planes 
than these; for, in such a plane must either lie two chords a, 
which leads to the curve 4°, or more than two chords /, which 
is the case for the points of 4*, but as for the latter only the plane 
through the tangent and the chord « comes into consideration ($ 2), 
the last possibility disappears and only the points of 4° are left. 
The double tangential planes of 2,, are therefore the planes x 
corresponiling to the points of k*; they envelope a developable A, of class 9. 
In order to find this number we look for all the double tangential 
planes passing through an arbitrary point B, of /*. Such a plane 
then must contain a chord of £* passing through B, intersecting 4°, 
and it can thus be obtained for instance by intersecting £® by the 
cubic cone projecting &* out of B, which furnishes 9 points of 
intersection, or inversely by intersecting £* by the cubie cone projecting 
k* out of the vertex 4,, which furnishes 12 points of intersection, 
of which three however coincide with ZB, and must be taken 
apart. If now we call A such a point of intersection lying on 4 
then really through this point passes one double tangential plane of 
2,, containing point 4,; so the class of the developable is nine. 
Through a point A of #* pass likewise 9 tangential planes of A, ; 
for one of these points A itself is the point from which start the 
two chords 5 of #*, in the eight other planes on the other hand 
the chords / start from an other point; from this ensues that through 
A pass altogether ten chords of 4‘ which start from the point of 
k* and which at the same time lie in the tangential planes of A, 
corresponding to those points; the locus of those chords is a surface 
2” of order twenty for which hk? is a tenfold curve. 
For, an arbitrary chord of #° meets in each of its 2 points 
of intersection with 4” ten generatrices of the scroll to be found, 
and is intersected outside #* by no chords of 4°. 
In a tangential plane of 4, lie also two chords 5 intersecting 4%, 
viz. in point A to which that tangential plane corresponds; Jet us 
also ask after the locus of these chords 5. Through each point of 4° 
pass two, through each point of 4 nine, because (see above) the 
cubic cone projecting /* out of that point is intersected by Z* in 
nine points; let us now determine the points of intersection of the 
scroll to be found with a chord 4, of 4*, then of these in each of the 
two points of intersection of 6, with #' lie nine united. If further- 
