1454 
through this nodal curve cut each other. Through a point P of hv 
pass two edges h,,h,, one lying on one sheet, the other on the other; 
the tangent plane in P at one sheet contains therefore b,, and the 
tangent ¢ in P at x’, and the tangent plane of the other the lines 
b, and ¢t. Now, however, h,,b, and ¢ lie in one plane ; i each point 
of ke the two sheets have consequently the same tangent plane. More 
may be said, however, viz. that the two sheets osculate each other 
along the whole curve kv. Let us namely suppose the normal plane 
of A” brought in P, and this plane intersected with £2, we shall 
then see two curves having the same vertical tangent in Pand being 
each other’s reflected image with regard to the normal n of £” lying 
in the base 8. The circle of curvature in P of one curve has its 
centre on 7, but as this circle is its own reflected image, it is at 
the same time circle of curvature of the other curve, from which 
it ensues that both curves osculate in P. And it may be further 
observed, as to the situation of the two sheets osculating along k’ 
that, at least in the neighbourhood of 4%, both must lie on the con- 
vex side of the cylinder which projects £” out of point Z,. 
In a node D of kv meet + sheets of £2, intersecting each other 
in pairs in 6 branches of the complete nodal curve of {2; two of 
them belong, however, to 4”, so that 4 remain belonging to the rest 
nodal curve, which are in pairs each others reflected image with 
regard to B and have all in D the same (vertical) tangent. As a 
twisted curve that has a vertical tangent in D projects itself on 8 
as a plane curve with a cusp in D, and the 4 branches of the 
nodal eurve lie in pairs symmetrically with regard to 8, the pro- 
jection of the rest nodal curve on B in D will show 2 cusps, both 
lying in that part of the plane from which the convex side of the 
two branches of k* is to be seen. Each of the 4 branches of the 
projection of the rest-nodal curve, meeting in J, is locus of points 
from where two equally long tangents may be drawn at /#, and 
these tangents always touch at both branches, not at one and the 
same branch (from which the number 4 of the branches may be 
easily deduced); if namely two equally long tangents are to touch 
at the same branch, the two sheets of £2, which pass through that 
branch, and which as we saw above osculate each other along that 
branch, must have another intersection in common, and this is only 
the case, as we shall see, in the neighbourhood of the so-called 
vertices of kv’, and these vertices are generally not situated in the 
immediate neighbourhood of the nodes. 
Let us now investigate the influence of the cusps of 4”. A cusp 
K causes in {2 2 cuspidal edges, one for each sheet, and lying in 
