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.

342 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
x3 can exist. If x2z2 exists also, then a + c = 0, d = 3a. The differential equation
is now
fxJ - 3yfy - Zfz + 3wfw = 0,
whence f- (x3y, z3, xz),
so that the most general form of the surface is
(X, 3w, o 2z2, y2 2, yzw )l=........................(10).
Let the terms xSy, xz3 co-exist, then b=-3a, a=-3c, d =- 7c; and the
differential equation is
- 3xfx + 9yfy + zfz - 7wf,l = 0,
whence f= b (x3y, xz3, X7W3),
and the only possible quartic terms are x3y, xz3, xyzw, so that the surface degenerates.
If the terms x3y, xyzw co-exist, then we get the surface (10) again. The cases
thus considered and those obtained by a mere interchange of variables exhaust all
possibilities, if a term such as x3y exists.
If no term cubic in any one variable exists, then the possible terms to be
considered are of the two types x2y2, xyz'w. If only one or no term of the former
type exists the surface degenerates; if terrns such as x2y2, x2z2 co-exist, then b=c;
if x2y2, z2W2 co-exist we revert to the case of (10).
We have thus considered all possible cases.
~ 4. TABULAR STATEMENT OF RESULTS.
The preceding analysis shews that if we exclude conical and degenerate surfaces,
there are five and only five types of quartic surfaces, given by equations (6), (7), (8),
(9) and (10), which satisfy Picard's differential equation, and are not pricma facie rational
surfaces. Surfaces which can be obtained from one another by linear transformation of
the coordinates are of course not counted as distinct.
After making some slight changes of notation with a view to greater uniformity,
arranging the surfaces in the order (9), (8), (7), (6), (10), and adding for convenience the
corresponding values of' 0, 02, 02, 0, we get the following table: