07 
pair lie at infinitesimal distance without intersecting each other, 6 
on the other hand indicates that they intersect each other without 
coinciding; the combination so therefore indicatesthe number of pairs 
the two components of which lie at infinitesimal distance and cut each 
other at the same time. This can take place in our case as follows. 
We know that the double tangential planes of /* are simply the 
tangential planes of the four quadratic cones cutting each other in 
ht; k° has with these four cones twenty-four points in common and 
through such a point pass evidently two generatrices satisfying 
the condition eo and forming together one pair satisfying this con- 
dition. These generatrices are the torsal lines of @** and their points 
of intersection with k* are the cusps. The surface 2** contains however 
also twenty double generatrices, viz. the common chords of 4° and 
k*, and these too must evidently be regarded as satisfying the indi- 
eated condition; the number eo is therefore = 20 + 24 — 44. 
The symbol eg indicates the number of pairs of rays which 
coincide and where g (or 4, which is of course the same) intersects 
a given line; now that given line intersects the surface in twenty- 
four points : so ey is twenty-four. We thus find: 
a es = EO JA tg == 44 gj 48 = 92, 
SO 
EB — 46. 
The symbol > indicates the condition that the two rays of a pair 
intersect a ray of a given pencil, thus the symbol eg indicates the 
condition that those two rays lie moreover at infinitesimal distance 
without intersecting each other; so the quantity e8 indicates in our 
case evidently exactly the class of a plane section of 27*. If now 
we remember that such a section contains in general no cusps, we 
then find for the number of double points : 
2d = 24. 23—46 = 552—46 = 506, 
d= 208. 
Now we know of these 253 double points the following: 1. the 
three points of intersection with “4°; 2. the four points of intersec- 
tion with &*, each of which is a ninefold point and therefore absorbs 
4.9.8—=36 double points; 3. the points of intersection with the 
twenty double generatrices, so together’ 3 + 4.36 + 20 = 167; 
the order of the new nodal curve is therefore 253 — 167 = 86. 
A plane curve of order twenty-four can possess at most $ . 23. 22 
= 253 double points, just the number of our case: 2° ús therefore 
a rational surface. 
