32 
ON THE SINGULARITIES OF SURFACES. 
[106 
plane is a stationary plane, as well of the flecnode-develope as of the spinode- 
develope, and it would at first sight appear that it must be also a stationary 
tangent plane of the node-couple-develope. But this is not so; the node-with-node- 
planes envelope, not the node-couple-develope, but the node-couple-develope twice 
repeated: the tacnode-plane is in a sense a stationary plane on such duplicate 
developable, but not in any manner on the single developable. The tacnode-plane is 
an ordinary tangent plane of the node-couple-develope. 
Consider now a spinode-with-node, which we have seen is a point of intersection 
of the spinode-curve and node-couple-curve. The tangent plane at a consecutive point 
along the spinode-with-node-tangent, is identical with the spinode-with-node-plane; the 
curve of intersection of the tangent plane at such consecutive point has therefore a 
node at the node-with-spinode, or the tangent plane in question is a node-couple- 
plane, and the point of contact is a point on the node-couple-curve. Consequently 
the spinode-with-node-tangent touches the node-couple-curve, and thence also the 
spinode-with-node-plane is a stationary tangent plane of the node-couple-develope. 
It should be remarked that no circumscribed developable can have a stationary 
tangent plane except the tangent planes at the points where the curve of contact 
meets the spinode-curve, and any one of these planes is only a stationary plane 
when the curve of contact touches the spinode-tangent; and that the node-couple- 
curve and the flecnode-curve do not intersect the spinode-curve except in the points 
which have been discussed. 
Recapitulating, the node-couple-curve and the spinode-curve touch at the tacnodes, 
and intersect at the spinode-with-nodes: moreover, the tacnode-planes are stationary 
planes of the spinode-develope, and the spinode-with-node-planes are stationary planes 
of the node-couple-develope. Besides this, the two curves are touched at the tacnodes 
by the flecnode-curve, and the tacnode-planes are stationary planes of the flecnode- 
develope.