106] 
ON THE SINGULARITIES OF SURFACES. 
31 
with-node-couple,” and the tangents to the curve of intersection at such nodes are 
“ node-with-node-couple-tangents.” The terms “ node-couple-with-node,” “ node-couple-with- 
node-tangent,” might be made use of if necessary. 
It should be remarked that the oscnodes lie on the flecnode-curve, as do also 
the fleflecnodes; these latter points are real double points of the flecnode-curve. The 
tacnodes are points of intersection and (what will appear in the sequel) points of 
contact of the flecnode-curve, the spinode-curve, and the node-couple-curve. The spinode- 
with-nodes are points of intersection of the spinode-curve and node-couple-curve, and 
the flecnode-with-nodes are points of intersection of the flecnode-curve and node-couple- 
curve ; the node-with-node-eouples are real double points (entering in triplets) of the 
node-couple-curve. 
Consider for a moment an arbitrary curve on the surface; the locus of the node 
tangents at the different points of this curve is in general a skew surface, which 
may however, in cases to be presently considered, degenerate in different ways. 
Reverting now to the flecnode-curve, it may be shown that the singular flecnode- 
tangent coincides with the tangent of the flecnode-curve. For consider on a surface 
two consecutive points such that the line joining them meets the surface in two 
points consecutive to the first-mentioned two points. The line meets the surface in 
four consecutive points, it is therefore a singular flecnode-tangent; each of the first- 
mentioned two points must be on the flecnode-curve, or the singular flecnode-tangent 
touches the flecnode-curve. The two flecnode-tangents are by a preceding observation 
conjugate tangents. It follows that the skew surface, locus of the flecnode-tangents, 
breaks up into two surfaces, each of which is a developable, viz. the locus of the 
singular flecnode-tangents is the developable having the flecnode-curve for its edge of 
regression, and the locus of the ordinary flecnode-tangents is the flecnode-develope. 
Of course at the tacnode, the tacnode-tangent touches the flecnode-curve. 
Passing next to the spinode-curve, the spinode-plane and the tangent-plane at a 
consecutive point along the spinode-tangent are identical 1 , or their line of intersection 
is indeterminate. The spinode-tangent is therefore the conjugate tangent to any other 
tangent line at the spinode, and therefore to the tangent to the spinode-curve. It 
follows that the surface locus of the spinode-tangents degenerates into a developable 
surface twice repeated, viz. the spinode-develope. Consider the tacnode as two coin 
cident nodes; each of these nodes, by virtue of its constituting, in conj unction with 
the other, a tacnode, is on the spinode-curve; or, in other words, the tacnode-tangent 
touches the spinode-curve, and the same reasoning proves that it touches the node 
couple-curve. It has already been seen that the tacnode-tangent touches the flecnode- 
curve ; consequently the tacnode is a point, not of simple intersection only, but of 
contact, of the flecnode-curve, the spinode-curve, and the node-couple-curve. 
In virtue of the principle of the spinode-plane being identical with the tangent 
plane at a consecutive point along the spinode tangent, it appears that the tacnode- 
1 It must not be inferred that the tangent plane at such consecutive point is a spinode-plane; this is 
obviously not the case.