fore cut by 2u+2vr—4e—26 
267 
in common with 2, if P is an arbitrary point of the cuspidal 
edge; a 4 point can therefore only arise as the cuspidal edge is 
cut by an ordinary generatrix of 2; the number of these points we 
find therefore by cutting the cuspidal edge with the 3°¢ polar sur- 
face. This third polar surface now is of order 2u + 2r—4e—20—38, 
and as this order is odd, the surface will have to contain again 
this plane 8 with a view to the symmetry with regard to 3; what 
remains is of order 2u--2»—4e—2o—4, and will be called 7, Of 
the intersections of this surface 7, with the cuspidal edge, one more 
point, however, lies in 8, viz. in the cusp Kk, so that only 2u + 
+ 2y—4e—2o—5 remain that do not lie in @. According to 
the preceding § the line Z, A has namely in A 4 points in common 
with ,, consequently 3 with zr, + 38; of these one belongs to 9, 
» and with a view to the symmetry of zr, 
with regard to 2, they can only lie on a tangent of 7,; 7, therefore 
passes through A with one sheet. Mach cuspidal edge of 2 is there- 
so that 2 remain for 7 
5 ordinary generatrices, and these points 
are cusps of the rest nodal curve, while their image circles cut k? twice 
perpendicularly, of which once in the associated cusp. 
The latter is a matter of course; that, however, the points in 
question are cusps of the rest nodal curve follows at once from the 
consideration of the generatrices of 2, which lie close to the one 
that cuts the cuspidal edge; they form namely with each other a 
certain sheet of &, and this of course cuts the two sheets meeting 
in the cuspidal edge in a cusp; the cuspidal tangent lies then in 
the cuspidal tangent plane of @, that is to say in the vertical plane 
passing through the euspidal edge. /n the projection we find there- 
fore for the locus of the points of equal tangents at ke a cusp, lying 
on a cuspidal tangent; and the number of these points on one and 
the same cuspidal tangent of k* amounts to 2u 2v—4e— 26 — 5. 
We saw above that in a cusp P of the rest nodal curve, lying 
on a cuspidal edge of 2, the line 7, P has 4 points in common 
with 2, and consequently with 2,, which also contains the cuspidal 
edge, 2; the tangent plane in P at 7, passes therefore through the 
cuspidal edge and is vertical. The cuspidal tangent in P lies now, 
according to the above mentioned fact, in this vertical plane, from 
which it ensues that the nodal curve and 2, bave 3 coinciding 
points in common in each point P/. 
The complete number of intersections of the rest nodal curve and 
a, amounts to (2u-+2r—4e—2e6—2)d; if we now put apart from 
this all the groups of points summed up in this and the preceding §, 
the triple points of the rest nodal curve remain, or, more clearly 
