1459 
As to the »—2e—o real foci a peculiar phenomenon is to be 
observed here; through these points passes, as we have seen, one 
branch of the rest-nodal curve, and the tangents at those points are 
vertical, consequently real, so that not only the foei themselves, but 
also the points infinitely near to them, therefore whole branches 
passing through those points, must be real, and consequently must 
have real projections on 8. Now it is a matter of course (think for 
instance of the conics) that neither the foei themselves, nor neigh- 
bouring points may be centres of circles cutting £” twice really, 
so that the branches of the nodal curve passing through the real 
foci are parasitic branches of the nodal curve, and there is nothing 
particular in this after all, for parasitic branches of the nodal curve 
separated from the ‘active’ parts by pinch-points, are met with 
already in the simplest ruled surfaces, as the wedge of Warris, the 
cubic ruled surfaces, the surface of normals, ete.; the peculiarity 
in our case is that the pineh-points are lying at infinity, and so the 
branches of the nodal curve passing through the foci nowhere reach 
the surface in fact. 
That this is correct indeed is easy to control on the parabola and 
the ellipse. For the parabola y? = 2 pw the tangent is y’y = p (t° + 2), 
and consequently the abscissa of the intersection with the v-axis : 
v= — ee’, while the distance from this point to the point of contact 
amounts to: V4a’? + y’2, orV 4a’? + 2pe’; if this distance is extended 
vertically upwards and downwards in the intersection of the tangent 
with the z-axis, 2 points of the nodal curve are evidently found, so 
that the equation of that curve becomes : 
2 == 4 2? — 2 px. 
If the origin is removed along the axis of the parabola over a 
distance of 4 p, so that it gets to lie half way between the vertex 
and the focus, and 2’ becomes = « — 4 p, the equation becomes : 
dp =p’, 
and this is an hyperbola cutting the plane 3 in the points a! = + 4 p, 
i.e. in the vertex and the focus of the parabola. But it is evident 
that only the branch passing through the vertex is really lying on 
the surface, whereas the one passing through the focus is parasitic 
as far as it extends. 
Of further importance is the observation that the directions of the 
asymptotes of the hyperbola are determined by the relation ——= + 2, 
ymp YI y <j ae 
so that half the asymptotic angle is greater than 45°; if therefore a 
point moves along the curve towards infinity, the associated circle 
94 
Proceedings Royal Acad. Amsterdam. Vol. XVIII 
