ON THE SINGULARITIES OF SURFACES. 
29 
106] 
the node. We may if we please use the term “nodes of a surface,” “node-planes of 
a surface,” as synonymous with the points and tangent planes of a surface. And it 
will be convenient also to use the word node-tangents to denote the tangents to the 
curve of intersection at the node ; it may be noticed here that the node-tangents 
are conjugate tangents of the surface. 
Next let two conditions be imposed upon the plane : there are three distinct 
cases to be considered. 
First, the curve of intersection may have a flecnode. The plane (which is of 
course still a tangent plane at the flecnode) may be termed a flecnode-plane ; the 
flecnodes are singular points on the surface lying on a curve which may be termed 
the “ flecnode-curve 1 ,” and the flecnode-planes give rise to a developable which may 
be termed the flecnode-develope. The “ flecnode-tangents ” are the tangents to the 
curve of intersection at the flecnode ; the tangent to the inflected branch may be 
termed the “ singular flecnode-tangent,” and the tangent to the other branch the 
“ ordinary flecnode-tangent.” 
Secondly, the curve of intersection may have a spinode. The plane (which is of 
course still a tangent plane at the spinode) may be termed a spinode-plane ; the 
spinodes are singular points on the surface lying on a curve which may be termed 
the “ spinode-curve 2 .” And the spinode-planes give rise to a developable which may 
be termed the “ spinode-develope.” Also the “ spinode-tangent ” is the tangent to the 
curve of intersection at the spinode. 
Thirdly, the curve of intersection may have two nodes, or what may be termed 
a “ node-couple.” The plane (which is a tangent plane at each of the nodes and 
therefore a double tangent plane) may be also termed a “ node-couple-plane.” The 
node-couples are pairs of singular points on the surface lying in a curve which may 
be termed the “node-couple-curve,” and the node-couple-planes give rise to a deve 
lopable which may be termed the “ node-couple-develope.” The tangents to the curve 
of intersection at the two nodes of a node-couple might, if the term were required, 
be termed the “ node-couple-tangents.” Also one of the nodes of a node-couple may 
be termed a “ node-with-node,” and the tangents to the curve of intersection at such 
point will be the “ node-with-node-tangents.” 
1 The flecnode-curve, defined as the locus of the points through which can be drawn a line meeting the surface 
in four consecutive points, was, so far as I am aware, first noticed in Mr Salmon’s paper “ On the Triple 
Tangent Planes of a Surface of the Third Order ” (Journal, t. iv. [1849], pp. 252—260), where Mr Salmon, 
among other things, shows that the order of the surface being «, the curve in question is the intersection of 
the surface with a surface of the order 11« - 24. 
a The notion of a spinode, considered as the point where the mdicatrix is a parabola (on which account 
the spinode has been termed a parabolic point) may be found in Dupin’s Développements de Géométrie : the 
most important step in the theory of these points is contained in Hesse’s memoir “ Ueber die Wendepuncte 
der Curven dritter Ordnung” (Grelle, t. xxviii. [1848], pp. 97—107), where it is shown that the spinode-curve 
is the curve of intersection of the surface supposed as before of the order «, with a certain surface of the 
order 4(n-2). See also Mr Salmon’s memoir “On the Condition that a Plane should touch a surface along 
a Curve Line” (Journal, t. hi. [1848], pp. 44—46).