266 
c= 4/404 4 + 2 (u— 2e — 20 — 1)}. (a — 2e — 20)0. 
The number of intersections of the rest nodal curve with 2, 
coinciding in Z, is therefore represented by the sum of the numbers 
a, hb and c. At infinity there lie, however, other intersections yet, 
viz. among others on #). This conic is for {2 a (r—o)-fold one and 
consequently for a, a (»—o—2)-fold curve, and contains according 
to §8 9—24 simple points of the rest nodal curve; the total amount 
of intersections on 4%, amounts therefore to (g—20) (r—o—2). 
As to the 25 points that are withdrawn from o@ they lie (ef. the 
example of the parabola in $ 5) on the straight lines Z, R,, and 
are intersections of the rest nodal curve with e,. As, however, 
two sheets of a, pass through each line Z, Rh, the number of 
intersections from this source becomes 4o. 
§ 11. We have now calculated how many intersections the rest 
nodal curve and a, have in 8, and how many they have in «, ; 
if there are more yet, they lie consequently neither in 8 nor 
in ¢,, and it is the nature of these points we really want to 
find out. As the rest nodal curve lies both on 2 and 2,, an inter- 
» and 
from this it ensues that the line connecting this point with Z, has 
ip this point 3 coinciding points in common with 2. This may in 
section of the rest nodal curve and a, lies on 2, a,, and a 
general happen as one of the 2 principal tangents of that point 
passes through Z,, but such a thing is excluded for the surface £2 
as we saw; if namely a vertical line should contain 3 infinitely 
near points of 2, 3 circles cutting 4 perpendicalarly might be 
drawn whose rays differ infinitely little, and this would only be 
possible if 4” possessed a tangent having contact in 4 points, which 
we have not supposed. Another possibility remains now, viz. that 
the points in question are triple points of @. Such a point arises 
when 3. different generatrices of 2 puss through the same point; 
it is in that case a triple point for the rest nodal curve, and its 
eyelographie image-circle will therefore cut /” thrice perpendicularly. 
There is, however, a third possibility yet, and this is concerned 
with cuspidal edges of 2. Through each cusp of 4% pass two cus- 
pidal edges of 2, and according to $ 9 each suchlike cuspidal edge 
is a hodal edge of 2,, and consequently also a simple straight line 
of a,, while the 3¢ polar surface does not contain it any more 
and so cuts in a certain number of points. Let us consider such an 
intersection P. This point lies on @ and on the Is, 2" and 3rd 
polar surface, from which it ensues that the line 7, P has in P 
4 coinciding points in common with &. Now has Z, P in P3 points 
