895 
however control this method here because we have to deal here 
with a cone instead of a regulus. The first polar surface of P; na- 
mely with respect to 2° is. a @° containing #° one time, and there- 
fore cutting 2° along £° counted twice and a residual curve of order 
24, so that the circumscribed cone at the vertex /P; is of order 24. 
Now this cone cuts the quadratic cone [P/] in 48 edges, so 48 edges 
of [P;] touch 2° and therefore #'*. The number of torsal lines of 
the first kind is thus indeed 48, and that this same number must 
now be found in general follows from the law of the permanency 
of the number. 
These numbers 6 and 48, as well as the number of points (namely 
40) which the nodal curve of 2° has in common with / can be 
controlled with the aid of the symmetrical correspondence of order 
70 existing between the planes 4 through / (§ 16). To the 140 double 
planes d belong, as we saw before, the planes through / and the 
nodal lines and those through / and the torsal lines of the first kind, 
together appearing there at a number of 54, but representing 60 
double planes. The nodal curve of @2*° has with / only the 4 points 
S; in common which however count for 10 each and which have 
the property that five of the six generatrices through each of those 
points lie in one plane; such a plane is thus undoubtedly a many- 
fold plane of the correspondence, the question is only how many 
single double planes it contains. Now there lie in the plane/7", e.g. 
9 generatrices through 7, cutting / in different points; through each 
of the last pass five other generatrices, and so we find so far 45 
planes conjugated to the plane {7 
Now we have moreover the plane through / and the 6" genera- 
trix: through S, (lying in vt); however by regarding, just as we: 
have done at the beginning of § 16, a plane 2 in the immediate 
vicinity of 77, and in which thus five generatrices cut each other 
nearly in: one point of / we can easily convince ourselves that 
this plane counts for 5 coinciding planes conjugated to /7,. To /7; 
are conjugated 45 +5—=50 planes not coinciding with 77 and 
thus 20 planes coinciding with /7,; i.e. just as in the general case 
a plane 4 through two generatrices cutting each other on / counts 
for two double planes, so here each plane /7’; containing five such 
generatrices counts for 5><4 double planes; so the four planes 
IT; represent 80 double planes, and they furnish with the 60 already 
found the 140 double planes as they ought to. 
As by the transition to a ray of the complex all numbers have 
remained unchanged, the surface 2° contains now again 58 torsal 
lines of the 2d kind; the 4>< 131 = 524 points of intersection 
