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.

340 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
IV. (ii). The auxiliary equations are:
dx dy dz dw
ax ay bz cw'
where 2a + b + c =0.
We may distinguish at once three sub-cases which may arise, viz.
(a) a=o, b=-c -0.
(/3) b=0, c=-2a0,.
(y) a$0, b#O, c O.
In SUB-CASE (a), three integrals are:
x = const., y = const., zw = const.,
so that f- ) (x, y, zw), and the quartic is
z2w2 + zw (x, )2 + (x, y)4=...................................(9).
In SUB-CASE (f), three integrals are:
z = const., yx = const., x2w = const.,
so that f= - (x2W, y/x,, z).
The only possible quartic terms are:
zw(x, y)2, z4
so that the surface degenerates.
In SUB-CASE (7) it is a little simpler to work directly with the corresponding
partial differentia] equation
ax- +~ay- + bz + cw -f=0,
x iay az awI
and to consider the possible terms in f.
Since the differential operator only alters the coefficient of any term, each term of
f must separately satisfy the differential equation.
We verify at once that the terms z4, w4, (x, y)4 cannot exist.
If a term of the type (x, y)3z exists, then 3a + b =0, whence a= c, contrary to
hypothesis; similarly no term of the type (x, y)3 w can exist. Similarly no terms of the
types (x, y)2z2, (x, y)2w2 can exist, as their existence would involve b=c.
Any term of the type (x, y)2 zw satisfies the equation.
If a term of the type (x, y) z exists, then a + 3b = 0, whence c= 5b, so that the
equation is
- 3 (xf~ + yfy) + zf, +.wfo = 0,