28 
[106 
106. 
ON THE SINGULARITIES OF SURFACES. 
[From the Cambridge and Dublin Mathematical Journal, vol. vn. (1852), pp. 166—171.] 
In the following paper, for symmetry of nomenclature and in order to avoid 
ambiguities, I shall, with reference to plane curves and in various phrases and 
compound words, use the term “ node ” as synonymous with double point, and the 
term “ spinode ” as synonymous with cusp. I shall, besides, have occasion to consider 
the several singularities which I call the “ flecnode,” the “ oscnode,” the “ fleflecnode,” 
and the “ tacnode: ” the flecnode is a double point which is a point of inflexion on 
one of the branches through it; the oscnode is a double point which is a point of 
osculation on one of the branches through it; the fleflecnode is a double point which 
is a point of inflexion on each of the branches through it; and the tacnode is a 
double point where two branches touch. And it may be proper to remark here, that 
a tacnode may be considered as a point resulting from the coincidence and amalga 
mation of two double points (and therefore equivalent to twelve points of inflexion); 
or, in a different point of view, [?] as a point uniting the characters of a spinode and 
a flecnode. I wish to call to mind here the definition of conjugate tangent lines of 
a surface, viz. that a tangent to the curve of contact of the surface with any 
circumscribed developable and the corresponding generating line of the developable, 
are conjugate tangents of the surface. 
Suppose, now, that an absolutely arbitrary surface of any order be intersected 
by a plane: the curve of intersection has not in general any singularities other than 
such as occur in a perfectly arbitrary curve of the same order; but as a plane can 
be made to satisfy one, two, or three conditions, the curve may be made to acquire 
singularities which do not occur in such absolutely arbitrary curve. 
Let a single condition only be imposed on the plane. We may suppose that 
the curve of intersection has a node; the plane is then a tangent plane and the 
node is the point of contact—of course any point on the surface may be taken for