259 
contact of £” and J, it has points of inflection, while the inflec- 
tional tangent coincides with / again; it is therefore clear that in 
/. a certain number of double tangents coincide. This number is 
easy to determine. Let us first imagine 2 of the o‘points of inflec- 
tion and let us observe that in each point of inflection 3 points of, 
the curve lie on a straight line, then it is clear that a double inflec- 
tional tangent absorbs 4 double tangents ; Lis, however, inflectional 
tangent for o points of inflection, consequently it absorbs 4 5 (¢—1). 4 
double tangents on account of this. 
Further is / (u—2e—2o)-times cuspidal tangent; as such it con- 
Qe —2o) (w—2e—2o—1) further donble tangents; 
tains therefore 3(« 
and finally each cusp may be combined with each point of inflec- 
tion,. which produces 2 double tangents every time, in consequence 
of the 3 points which lie infinitely near to each other in the point 
of inflection; to the two preceding numbers 2o(u— 2e— 20) must 
still be added. If the sum of these three numbers is subtracted 
from the number given higher up, we find for the number of double 
normals of /”: 
 (2ur— 2u0— 8u-+ v* — dweep — p+ 466 4+ 0? 4-30 — 0). 
To each of these two points of 4% harmonical with regard to 
4, and 3 are associated; and if we now subtract the double number 
of double normals from the number of points that ¢, outside 7, 
has in common with the rest nodal curve, as was given at the 
beginning of this §, we find that 
Zur - 2u0—A4yve + 4e0— 1 2v+ 8 + 6,4 Budd 
remain. 
Now we know already from the example of the parabola ($ 5) 
that points of this kind exist in fact; there we found 2, viz. the two 
points at infinity of the nodal curve of 2, and the number arrived at 
here really gives 2 for the parabola. These points lie on the line 
connecting Z, with the point of contact of the parabola and /, 
and form a necessary completion of a few numbers found in $ 
where we namely in § 7 considered tangents of equal length at 
different parabolical branches or at parabolical and hyperbolical 
branches, there we have of course also to consider tangents of equal 
pen 
d 
; 
length at one and the same parabolicai branch, and this we find 
here now ; the total number of these points amounts to 20, so that 
the remaining ones indicate parabolical branches of the rest nodal 
curve. 
§ 9. The order of the rest nodal curve of £ may moreover be 
Ai hg 
‘ 
