595 
ray s conjugated to 7; lies in 7,, but not in / 7;; so the points $; 
are 4.4.38 = 6-fold points for the nodal curve and others this curve 
ean evidently not have in common with /. So it has 24 points 
united in 4 sixfold points in common with /, and as there are in 
a plane « through / §.13.12 = 78 points not lying on / the order of 
the nodal curve now amounts to 24 + 78 = 102. The number of 
nodal points of a plane section of 2'* amounts thus now to 
102 +6+10=118, and from this ensues for the class 18.17 — 
—2118=70=e8; the formula eo = 2. eg — 2. eg furnishes there- 
fore «6 = 2,70 — 2.18 — 104 torsal lines of both kinds. 
The formula 
E=P III 
now again applied to determine the number of generatrices of the 
cone [P] touching #* and thus of the number of torsal lines of the 
first kind gives the following results. The plane of the condition p 
euts £ in 12 points; through each of these passes a generatrix of 
the cone cutting 2° besides in P/ in four points more; so the num- 
ber p is equal to 48, a and likewise g. The line of the condition g cuts 
the cone in two points and through each of these passes a genera- 
trix of that cone, on which lie besides #, five points of ks so g 
is = 2.20, and thus e= 2.48 — 2.20 = 56. Among these however 
are included the six nodal lines counted twice; the number of 
torsal lines of the first kind amounts thus to 56 — 2 x 6 = 44. 
To control this we again consider the first polar surface of P, 
with respect to 2°, a 2*® touching 2° in P; and passing through 
k*, The intersection with 2° consists therefore of 4° counted twice and 
a residual curve of order 30 — 2.38 = 24 which however is projected 
out of 7) by a cone of order 22 only, because 7; itself is a nodal 
point of that curve (for @° and @? touch each other in P)); this 
cone has with the cone [|/,| 44 generatrices in common, and 
these touch £"?. 
The number of torsal lines of the 2° kind of B amounts to 
104 — 6 — 44 — 54. 
The correspondence of the planes À through / is now of order 52 
with 104 double planes. For, in a plane 2 lie besides / thirteen ge- 
neratrices of 2'° and through each of the 13 points in which these 
cut 7 four others pass; so to each plane 44 x 13 = 52 others are con- 
jugated. The double planes are 1. the planes through the 44 torsal 
lines of the first kind; 2. tbe planes through the 6 nodal edges, 
each counted twice; 3. the 4 planes /7; each counted twelve times, 
because in each such like plane 4 generatrices pass through the point 
S; (comp. § 18); so we find 44 + 2.6 + 4.12 = 104 double planes, 
