725 
to which d corresponds as “branch plane” '); if we remember that in 
d, besides s, and s,, lie only twelve other generatrices of 2°, then 
to d are conjugated 12*%5+2<4=68 planes not coinciding 
with d. The two missing ones do coincide with J, so that dis really 
a double double plane. 
It is easy to see that the reasoning given here is literally applicable 
to the six double generatrices, but not to the torsal lines of the first kind, 
and much less to those of the second. The plane 4 through a torsal 
line of the first kind is, it is true, a double plane J, but only a 
single one, for besides that torsai line there are now in d still 13 
other generatrices of 7° (because namely the torsal plane does not 
pass through /), and through the pinchpoint pass four generatrices 
not lying in d; so to d are now conjugated 13 X 5 + 4 == 69 
planes, so that only one coincides with d. And as for the torsal lines 
of the second kind, these give no rise whatever to double planes, but 
only to branch planes. Let us assume again, as above, a plane A, in 
which lie two generatrices s,,s, which almost coincide, but in such 
a way, that their point of intersection lies at finite distance from 
[. Through Z, and LZ, pass again every time five generatrices not 
lying in 2, but now lying neither in the vicinity of s, nor of s,, and 
when 2 transforms itself into the plane through / and the torsal line 
of the second kind, those ten generatrices coincide two by two; so 
the torsal plane becomes a fivefold branch plane, but not a double plane. 
Let us now draw the cenclusion from these considerations. If we 
assume the double curve of 2” to have z points in common with /, 
then our correspondence contains wv + 6 (namely on account of the 
double generatrices) double planes counting twice, and 48 (on account 
of the torsal lines of the first kind, see $ 14) double planes counting 
once, so that the equation exists: 
2 (w +6) + 48 — 140 
out of which we find: z= AO: 
So the double curve of 2*° rests in 40 points on | and is there- 
fore of order 40 + 91 = 181. 
A plane section of @2*° contains however not only 131 double 
points, but 1831 + 6+ 15 = 152, viz. 6 on the generatrices and a 
sixtold- points ón /5~ so. it is of class 20 «19 — 2. SC 152 =—,76, ‘so 
that if we again apply the formula 
eO=—2.e8—2.89 
we must substitute for ¢8 the number 76; and as eg — 20, because 
the line of the condition g intersects 2° in 20 points, we find. 
1) Em. Weyr, “Beiträge zur Curvenlehre,” pp. 9, 10, 
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