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.

344 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
then leads to
x ' = ( ', ')........................................ (A
(B) Choosing as a ne\ variable y one of the factors of (x, y)32, we can write the
equation:
z2 (x, y)1 + zw (x, y)22 + w2y (x, y)1 = 0.
The quadric transformation:
x: y: z: w =x'z': y'z': y'w': z'w'
x' y' z'.: w= xz yz:yw:zw
then leads to
y' (x, y') + z' (', y') + (', y') = 0........................... (B').
(C) The coordinates can evidently be chosen so that the point x = y = z = O lies on
the surface; z is then a factor of (z, w)4, which may accordingly be written z (z, w)3.
The quadric transformation:
x: y: z: w = z' (x' + y'): y 'w': z2w'
': y:: ' = xw-yz: yz: 2 zw
then leads to
z' + 2x' (z', w')' + (z', w')= 0................................. (C').
(D) Changing the variables as in (C) and employing the quadric transformation:
x: y: z: = (x'z + y'2): y'w: z': w 2
': y': z': w' = 2xw -y: yz: z: zw
we get x"2' + 2' (z', W')2 + (', w)=......................
(E) The cubo-quartic transformation:
x: y: z: w = y/2z': 'w3: x'y'z'w': y'yw'2
xz: y z: z:w' = z2w: XW2: Xyz: Xyw
converts the surface into
ay'z'2 + bx'y' + cx'y'z' + dx'y' + ex'2z' = 0..........................(E').
The five surfaces (A')-(E') are cubic cones, which are in general non-singular, so that
each possesses an integral of the first kind. The birational transformation of such an
integral converts it into an integral of the first kind on the corresponding qiiartic surface.
Moreover, if the coefficients which occur in the equations are left arbitrary, the
five cones are perfectly general cubic cones, though they occupy special positions relatively to the coordinate planes. Hence we see that a quartic of any of the five types
can be birationally transformed-if necessary via a cubic cone-into a quartic belonging