From this simple example we may now draw important conclu- 
sions for the general case. Even then the sheets 1 and 2 cut each 
other on one side of 8 in a 45°-line, on the other in a curve, and 
the latter is completed by its image and a parasitic part into a 
curve with a node in A. With the sheets 1 and 2* it is in so far 
different that they eut each other both above and below 8 in branches 
of curves, both completed again by parasitic parts into a curve 
with a node in A, and finally the sheets 1* and 2 of this last curve 
produce moreover the image. Apart from the two cuspidal edges 
(45°-lines) therefore, the restnodal curve of Q possesses 3 nodes in 
each cusp of k,; and as of the three curves in question here one 
is its own image, whereas the two others are each other’s image, 
the projection of the restnodal curve in the neighbourhood of K will 
consist of 3 branches which all touch at the cuspidal tangent. This 
may again be easily perceived planimetrically. The projection 
of the two cuspidal edges is the cuspidal tangent of A, and the 
latter is the locus of the centres of all the circles which cut the 
two branches of 4” meeting in A perpendicularly in this point. The 
two branches of the rest-nodal curve, which stereometrically belong 
according to the considerations put down in the preceding §, to the 
cuspidal edges, and complete them into curves with a break in 
them, project themselves into a branch containing all the points out 
of which two equally long tangents at £” pass which are both 
turned away from A; the two other branches contain the points 
out of which one tangent of A is turned away from, the other 
turned towards A. 
Of the two 45°-lines passing through A’ we found in the preceding 
$ so to say every time only one half, but the other halves have 
their signification too. Let us viz. to that purpose consider a node 
D with a small knot while the nodal tangents almost coincide 
already. If we follow this small knot from the node to the node, 
we see the circle of curvature decrease at first, but afterwards 
increase; it has been a minimum in one point, and this point is 
for hk” a vertex, that is to say a point where the circle of curvature 
touches in 4 points; and it is easy to see now that from this point 
a branch of the projection of the rest-nodal curve must start; for, 
if we suppose the 4 points which the circle of curvature has in 
common with #7, infinitely near, and then call them 1, 2, 3, 4, 
there pass through the intersection of the lines 12 and 34 two 
tangents at kv, which at the same time touch the circle of curvature, 
and are therefore equally long. The two sheets of @ near a 
vertea of k* cut each other consequently along a with regard to B 
