506 
al 
more we make a plane £ to rotate round 6, then the chord 4, in that 
plane, which cuts 6, outside £*, describes a scroll having six points 
in common with %°; through each of these passes a chord 4, which 
cuts A? and b,; the scroll to be found is therefore a surface 2** of 
order 2 X 9+6= 24. It has kh’ as a nodal curve and k* as a ninefold 
curve. 
9 The surface 2*° found in the preceding $ possesses no other 
manifold curve than 4°. Each scroll of order contains namely a 
nodal curve which is cut by a generatrix in n—2 points, because a 
plane through a generatrix contains as residual section a curve of 
order n—1, and of the n—1 points of intersection of this curve 
with the generatrix only one acts as a point of contact, so that all the 
remaining ones are due to a nodal curve. Now a plane through a genera- 
trix of 2° contains a residual section of order nineteen with two 
ninefold points on /*; these together form eighteen points of inter- 
section of the generatrix with tbe nodal curve, so that the latter is 
complete with 4° only. On the other hand the surface contains twenty 
double generatrices, viz. the common chords of k* and k*, as is easy 
to see, and these same lines are double generatrices of 2**. 
The surface 2** contains besides the nodal curve 4° and the ninefold 
curve /* still a new nodal curve which is cut by each generatrix in 
five points: for, a plane through a generatrix contains a residual 
section of order twenty-three with two eightfold points on /* and 
a single point on 4°, forming together seventeen points; so the gene- 
ratrix must contain five points more of an other nodal curve. And 
indeed, if we make a plane to rotate round a generatrix 6,, it then 
possesses in each position still one chord 5, of #* not meeting +, 
on k*; this chord describes a regulus intersected by 4° in six points, 
of which one however coincides with the point of intersection of 0, 
and 4°; through the remaining five passes every time one generatrix 
of 2** meeting b, outside &? and 4*, thus in a point of the new 
nodal curve. 
We can find the order of this new nodal curve with the help of the 
theory of the permanency of the number. We conjugate an arbitrary 
generatrix of @2?* which we call g to all others which shall then 
be called /, and in this way we find oo pairs of lines gh to which 
we will apply in the first place Scnupert’s formula: 
60 2.88 -2. 6&9"). 
The letter ¢ indicates the condition that two rays g and h of a 
1) ScuuperT: “Kalkül der abz. Geom.”, p. 60, N°. 22, 
