30 
ON THE SINGULARITIES OF SURFACES. 
[106 
106] 
It is hardly necessary to remark that the flecnode-curve is not the edge of 
regression of the flecnode-develope, and the like remark applies m.m. to the spinode- 
curve and the node-couple curve. 
with-node-i 
“ node-wit! 
node-tange 
Finally, let three conditions be imposed upon the plane : there are six distinct 
cases to be considered, in each of which we have no longer curves and developes, 
but only singular points and singular tangent planes determinate in number. 
It sh 
the flefleci 
tacnodes i 
contact of 
First, the curve of intersection may have an oscnode. The plane (which continues 
a tangent plane at the oscnode) is an “ oscnode-plane.” The “ oscnode-tangents ” are 
the tangents to the curve of intersection at the oscnode ; the tangent to the 
osculating branch is the “ singular oscnode-tangent ; ” and the tangent to the other 
branch the “ordinary oscnode-tangent.” I • 
with-nodes 
the flecnoc 
curve; the 
node-coupl 
Consi( 
Secondly, the curve of intersection may have a fleflecnode. The plane (which 
continues a tangent plane at the fleflecnode) is a “ fleflecnode-plane.” The “ fleflec- 
node-tangents ” are the tangents to the curve of intersection at the fleflecnode. 
tangents ! 
may howei 
Rever 
Thirdly, the curve of intersection may have a tacnode. The plane (which 
continues a tangent plane at the tacnode) is a “ tacnode-plane.” The “ tacnode- 
tangent” is the tangent to the curve of intersection at the tacnode. 
tangent ce 
two conse 
points con 
four conse 
Fourthly, the curve of intersection may have a node and a flecnode, or what 
may be termed a node-and-flecnode. The plane (which is a tangent plane at the 
node and also at the flecnode, where it is obviously a flecnode-plane) is a “node-and- 
flecnode-plane.” The “ node-and-flecnode-tangents,” if the term were required, would be 
the tangents to the curve of intersection at the node and at the flecnode of the 
node-and-flecnode. The node of the node-and-flecnode may be distinguished as the 
node-with-flecnode, and the flecnode as the flecnode-with-node, and we have thus the 
terms “ node-with-flecnode-tangents,” “ flecnode-with-node-tangents,” “ singular flecnode- 
with-node-tangent,” and “ordinary flecnode-with-node-tangent.” 
mentioned 
touches tl 
conjugate 
breaks up 
singular fl 
regression, 
Of course 
Passin 
consecutive 
Fifthly, the curve of intersection may have a node and also a spinode, or what 
may be termed a “ node-and-spinode.” The plane (which is a tangent plane at the 
node, and is also a tangent plane at the spinode, where it is obviously a spinode-plane) 
is a “ node-and-spinode-plane.” The node-and-spinode-tangents, if the term were 
required, would be the tangents at the node and the tangent at the spinode of the 
node-and-spinode to the curve of intersection. The node of the node-and-spinode 
may be distinguished as the “ node-with-spinode,” and the spinode as the “ spinode- 
with-node,” and we have thus the terms “ node-with-spinode-tangent,” “ spinode-with-node- 
tangent.” 
is indeterr 
tangent li 
follows thi 
surface tw 
cident noc 
the other, 
touches tl 
couple-cur 
curve; coi 
contact, of 
Sixthly, the curve of intersection may have three nodes, or what may be termed 
a “ node-triplet.” The plane (which is a triple tangent plane touching the surface at 
each of the nodes) is a “ node-triplet-plane.” The “ node-triplet-tangents,” if the term 
were required, would be the tangents to the curve of intersection at the nodes of 
the node-triplet. Each node of the node-triplet may be distinguished as a “ node- 
In vi; 
plane at < 
1 It mus 
obviously not