ON A SEXTIC TOESE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
pp. 229—235.] 
The torse having for its edge of regression or cuspidal edge the curve defined by 
the equations x = cos <f>, y = sin (f>, z = cos 2(f), is an interesting and convenient one for 
the construction of a model, and it is here considered partly from that point of view. 
The edge is a quadriquadric curve, the intersection of the cylinder a? + y 2 = 1 with 
the parabolic hyperboloid z — x? — y 2 ) the cylinder regarded as a cone having its vertex 
at infinity on the line x = 0, y — 0, viz. the vertex is on the hyperboloid, or the curve 
is a nodal quadriquadric (the node being thus an isolated point at infinity on the line 
in question), and the torse is consequently of the order 8 — 2, =6, viz. it is a sextic 
torse. 
The edge is a bent oval situate on the cylinder x* + y 2 = 1, such that, regarding </> 
as the azimuth (or angle measured along the circular base from its intersection with 
the axis of x), the altitude £ is given by the equation z = cos 2(f> ; viz. there are in 
the plane xz, or, say in the planes xz, x'z, two maxima altitudes z— 1, and in the 
plane yz, or, say in the planes yz and y'z, two minima altitudes z = — 1. The sections 
by these principal planes are, as is seen at once, nodal curves on the surface ; they 
2 
are, in fact, the cubic curves z = 3 , viz. here as x increases from +1 to + oo, 
x- 
z increases from the before-mentioned value 1 to 3, and z = — 3 + —, viz. as y increases 
y 2 
from ±1 to +oo, z decreases from the before-mentioned value — 1 to — 3. The two 
half-sheets (which meet in the cuspidal edge) intersect each other along these nodal 
lines, in suchwise that the section of the surface by any axial plane (plane through 
the line x = 0, y = 0) is a curve having a cusp on the cuspidal edge, and such that 
when the axial plane coincides with either of the principal planes x = 0, y — 0, the