1458 
symmetrical curve, which in the vertex itself has a vertical tangent. 
And it will be elear now without further demonstration that if 
the node D passes into a cusp A, the vertex of the small knot gets 
to lie in A, and the new branch of the rest-nodal curve just found 
passes into the two halves not yet accounted for of the 45°-lines 
passing through A. 
$ 5. The points of inflexion of 4”, as is easy to understand, are 
not directly connected with the rest-nodal curve. The vertical plane 
passing through an inflexional tangent contains two systems of 
generatrices, mutually parallel and with regard to 2 symmetrical, 
lying infinitely near and they are evidently torsal-lines of 2, but they are 
in no way connected with the nodal curve; on the other hand there 
are in 3 two groups of points that do belong to the rest-nodal curve, 
and which we have not yet discussed in the preceding §. According 
to § 2 there pass through each of the two absolute points at infinity 
p—Je—o tangents at £? and each of them meets /£” except in the 
point of contact and the cyclic point in question, moreover in 
u—e—2 other points; through the point of contact passes no other 
generatrix but the isotropic tangent itself, counted twice, so that 
this point does not belong to the rest-nodal curve (it is a pinch- 
point of £2, of course an imaginary one, and along the isotropic 
tangent two sheets of the surface pass into each other); in each of 
the «—s—2 other points, however, the sheet, to which that isotropic 
„tangent belongs, cuts the two sheets which pass already through 
those other points, so that two branches of the rest-nodal curve 
appear, which in such a point pass through 3 with a vertical tangent; 
so we find the following result. ln each of the 2(u—e—2)(r—2e—o) 
points which the tangents out of the two isotropic points of B have, 
besides these points and the points of contact, moreover in common 
with kv, passes the rest-nodal curve with two branches through 8, 
which branches osculate each other along a vertical tangent. These 
points are of course all imaginary. 
But we have further to consider the points in which the y—2e—o 
tangents out of the isotropic point /,, ent the tangents out of /,,; 
these points amount to (»—2«—o)’, and among them are »y—2s—o 
real ones; they are the so-called foci of £“'). Through each of these 
points pass two single sheets of the surface, and consequently passes 
one single branch of the vest-nodal curve; so we find: the (p—2e—o)* 
foci of ke are single intersections of the rest-nodal curve with B. 
1) Cf. Ann. Gykl. p. 25. 
