Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

346 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
the surface can be transformed by linear substitution (x'= x + iy, y'= x- iy) into the
general quartic surface of revolution.
The birational transformation ernployed in ~ 5 establishes a one-one correspondence
between points on the conics and points on the generators of the cubic cone.
The surface (B) is the well-known quartic scroll with two non-intersecting double
lines, which is Cayley's first* and Cremona's eleventht species of quartic scroll.
The surface (C) is Cayley's fourth and Cremona's twelfth species of quartic scroll,
and is the limiting form assumed by the preceding surface, when the two double lines
coincide without cutting one another, thus giving rise to the higher singularity sometimes called a tacnodal line+.
The generators of the surfaces (B) and (C) correspond to the generators of the
cones into which the surfaces can be transformed.
The surface (D) has a double point at y=z = = 0, which is for some purposes
at least equivalent to two tacnodes, as defined above; and the surface can be regarded
as a limiting form of the surface (A) when the two tacnodes coincide. A section by
a plane through = w =O breaks up into two conics which have contact of the third
order at the singular point. This singularity can be defined-in a form applicable to
a surface of any order-as a uniplanar double point such that a section by an arbitrary
plane through some fixed tangent line at the point has two branches meeting one
another in four points at the singular point. This property implies that in the case of a
quartic the section breaks up into two conics. As far as I am aware neither this
singularity nor the surface has hitherto received any attention.
As before the conics correspond to the generators of the cubic cone.
It may be observed that though the surfaces (C) and (D) can be regarded, from
a geometrical point of view, as limiting cases of Poincaré's surfaces (A) and (B), they are
not analytically special cases of them, that is, the equations (C) and (D) cannot be
derived from (A) and (B) by giving special values to the coefficients.
The remaining surface (E) does not appear to have been studied hitherto. It has two
precisely similar uniplanar points of a rather complicated character, which can be stated
in a form applicable to a surface of any order somewhat as follows. The section by
the plane tangent at the point has a triple point, there, as always happens with a
uniplanar or biplanar point; but in addition the three branches at the triple point
coincide in direction, and if we call their common tangent the singular tangent line,
this line meets the surface not merely in 4 but in 5 coincident points: thus in the
quartic case this tangent line lies wholly on the surface. At an ordinary uniplanar point
a section by a plane through a singular tangent line has a tacnode (equivalent to two
* "A Second Memoir on Skew Surfaces, otherwise + "Sulle superficie gobbe di quarto grado," Mem. di
Scrolls," Phil. Trans., t. 154 (1860): Coll. Math1. Papers, Bologna, ser. II. t. vImI. (1868).
t. v. no. 340. + Salmon's Geometry of Three Dimensions, ~ 556.