of equal tangents at kv. The total number of these points amounts 
to 2o0(7—1), and to the same two points of contact R,,, Re, of 
kv and l belong 4 of those points. If they are considered as centres 
of circles twice cutting 4” perpendicularly, the circle itself always 
coincides with */,, and though / does not eut 4” in Ry, R 
co? 
but touches it, there can be no objection, for the line /, encloses 
with itself any arbitrary angle. Even planimetrically it is clear that 
to two points of contact with 2, belong 4 simple infinite branches 
of the locus of equal tangents; if we viz. imagine 2 parabolic branches 
of kk”, out of one point or another go 2 tangents at each of them, 
so that for the equality of 2 of those tangents, touching at different 
branches, there are 4 possibilities; in this way 4 simple infinite 
Ze? 
branches of the curve arise. 
The branches of the 3 kind finally arise from the combination 
of a simple intersection S, of 4’ and /, with a point of contact 
R., (§ 6); they arise in groups of 4, are associated to each other in 
pairs in the involutory collineation Z, 3, and have all 4 only one 
tangent in Z,, viz. apparently the line Z, 5, itself; the consequence 
of this is that through S, pass 2 branches of the locus of equal 
tangents, that is to say 2 branches for each point /, consequently 
25 together. So: the w~—2e—2o6 simple intersections of hv and L, are 
for the locus of the points of equal tangents 26-fold points. 
This result as well allows of being verified planimetrically. 
Let us imagine a hyperbolic and a parabolic branch of 4”, out of 
a point of 3 near the asymptote passes then one tangent lying very 
close to that asymptote, while two tangents touch at the parabolic 
branch ; so there are 2 possibilities for the equality of those tangents, 
and consequently 2 branches of the locus of the points of equal 
tangents go in the direction of the asymptote towards intinity. Even, 
as ZS, is a torsal line of 2 the plane of osculation in Z, will 
coincide with the torsal plane, and consequently the tangent in ‚5, 
at each branch of the locus of the points of equal tangents with the 
asymptote of 4’. We may therefore add to the preceding : all branches 
of the locus of the points of equal tangents passing through a pout 
S, of ke, have in this point the asymptote of ke as a tangent. 
§ 8. In § 6 we have determined the number of intersections that 
an arbitrary plane passing through Z, has apart from Z, in common 
with the rest nodal curve of 2, and we have determined from it 
the order d* of the locus of the points of equal tangents at /” ; for 
the plane ¢,, however, the calculation is somewhat different, as 
several branches passing through Z, touch the plane ¢,,. The branches 
17 
Proceedings Royal Acad. Amsterdam, Vol. XIX. 
