1455 
the vertical plane passing through the cuspidal tangent, and of 
course at angles of 45° with regard to 3; the acute edges point 
both to the same side as the acute point of A. In order to discover 
now the conduct of the rest nodal curve of £2 in the neighbourhood 
of A, we just replace the cusp by a node D with a little loop and 
then intersect {2 with a plane lying in the neighbourhood of D 
(but not on the side of the loop), and for convenience, sake thought 
vertical. Let us suppose 4” in the environment of D exactly drawn, 
the intersection of £2 with the plane is also easily and sufficiently 
exactly to be constructed; two branches 1 and 1* are found lying 
symmetrically with regard to ps, and also two others, 2 and 2*. 
The branches 1 and 2 intersect each other in 2 points, 1* and 2% 
in those symmetrical with regard to 8, and when the plane of 
intersection is moved these four points describe two with regard to 
8 symmetrical branches of the nodal curve, which project themselves 
on 8 in one curve with a cusp in D, as has been explained above. 
And the same holds good wiih regard to the branches 1 and 2%, 
and 1* and 2 respectively. 
If, however, the node passes into a cusp, the branches 1 and 2 
join (and 1* and 2* symmetrically) into a cusp, lying on one of 
the two 45° lines passing through A, mentioned above, whereas the 
second intersection remains arbitrary; by removal of the plane of 
intersection in the direction of A, one intersection describes the 
45°-line, however, no farther than A, the other a curve ending in 
AK, and that, as a simple investigation will teach, with an arbitrary 
inclination with regard to 3; the continuous curve passing through 
D, which had a vertical tangent in D, has therefore passed into a 
curve showing a break in A, and composed of a true curve and a 
piece of a 45-line. And the branches 15, 2*, produce, it is true, of 
that curve the image, but as the tangent in A, as we shall see, is 
generally speaking not vertical, a break remains in existence here 
as well. 
As, however, £2 is algebraic, every discontinuity is seemingly 
done away with again, and this happens here whereas the curve 
with vertical tangent in the node passes, in the case of the cusp 
into a curve with a node in A, and of which two branches, which 
are each other’s image with regard to 5, are active, the two others 
parasitic. 
Let us take as a simple example the curve 7? = a*, which has 
the advantage of possessing an axis of symmetry, so that one of 
the branches of the nodal curve passing through A (or more exactly 
two) gets to be situated in the plane of symmetry of £2. By means 
