I take account of conical and biplanar nodes, or, as I call them, cnicnodes, and binodes; of pinch-points on the nodal curve; and of close-points and off-points on the cuspidal curve: viz. I assume that there are
C, cnicnodes,
B, binodes,
j, pinch-points,
𝜒, close points,
𝜃, off-points,
deferring for the present the explanation of these singularities. The same letters, accented, refer to the reciprocal singularities. Or using trope
as the reciprocal term to node, these will be
C', cnictropes,
B', bitropes,
j', pinch-planes,
𝜒', close-planes,
𝜃', off-planes;
but these present themselves, not in the equations above referred to, but in the reciprocal equations.