ON THE CUSP OF THE SECOND KIND OK NODECUSP. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vi. (1864), 
pp. 74, 75.] 
The so-called cusp of the second kind or ramphoid cusp, is not an ordinary 
singularity of plane curves, but it is a singularity of a higher order. It is however 
particularly considered in Plücker’s Theorie der Analytischen Gurven, 1839; and it is 
there, not only in the analytical discussion of the singularities of plane curves, but 
in the author’s theory of the generation of a curve, considered as described and 
enveloped by a point moving along a line which at the same time rotates round the 
point; when the motion along the line vanishes, we have a cusp; when the 
motion round the point vanishes, we have an inflexion; when the two motions vanish 
together, we have a cusp of the second kind, which thus presents itself as a singularity 
uniting the characters of a cusp or stationary point, and an inflexion or stationary 
tangent: (I remark in passing that in this explanation it is not clear what is the 
independent variable wherewith the motions are compared). But there is another point 
of view from which the singularity in question may be considered, viz., it may be 
regarded as a singularity arising from the union and amalgamation of a cusp, and a 
double point or node; in fact, in the figure, which represents a curve having a cusp 
and also a node, we have only to imagine the node approaching nearer and nearer to 
and ultimately coinciding with the cusp, and it will be at once seen that the point 
will become a cusp of the second kind; or as it might properly, with reference to 
c. v. 34