54 
ON THE QUARTIC SURFACES REPRESENTED BY 
[647 
For each of the above-mentioned four values of A, the quadric cone breaks up 
into a plane-pair; each plane of the plane-pair is thus a “trope” or plane touching 
the surface along a conic; viz. this is the conic passing through the intersection of 
the plane (or say of an axis) with the nodal line and through four nodes of the 
surface. We have thus 8 tropes, intersecting in pairs in the four axes (and the inter 
section of each axis with the nodal line being a pinch-point). Moreover, joining the 
nodes in pairs, we have four rays, each meeting the nodal line, the plane through it 
and the nodal line being a pinch-plane; this is illustrated in the figure. 
As to the pinch-planes and pinch-points, remark first that a plane through the 
nodal line is in general a bitangent plane, its two points of contact being the points 
where the conic in such plane meets the nodal line. When the two points of contact 
come to coincide, the plane is a pinch-plane; viz. this happens when the plane passes 
through a ray, the conic being then the ray twice repeated. And secondly, at a point 
on the nodal line there are in general two tangent planes, viz. these are the tangent 
planes to the quadric cone belonging to such point; when the two tangent-planes 
come to coincide the point is a pinch-point, and this happens when the point is the 
intersection of the nodal line with an axis, for then (the quadric cone breaking up 
into the two tropes through the axis) the two tangent planes become the plane 
through the axis taken twice. 
Each section through the nodal line is a conic, and the polar of the nodal line 
in regard to this conic is a point; the locus of this point (for different sections 
through the nodal line) is a right line which may be called simply the “polar.” To 
prove this, considering the section by the plane y = 6x, we have to find the pole of 
the line % = 0 in regard to the conic 
(a', b', c, f', g', h'Qlx, m, n) 2 = 0;