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.

338 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
The general quartic of this type is
(z - 2yw)2 + 2 (2 - 2yw) (x, )2 + (, y)4 = 0........................ (6),
where (x, y)n denotes an arbitrary quantic of order n.
I. (iii). The auxiliary equations are:
dx dy_ dz _ dw
O x O z
leading to x = const., z = const., yz - xw = const., and
f- 4 (, z, yz - xw).
The general quartic of this type is
(yz - xw) + 2 (yz - xw) (x, )2 + (x, )4= O........................... (7).
I. (iv). The auxiliary equations are:
dx dy dz dw
x 0 O'
leading to the cone
ff,,(x æ, Z )= O.
II. (i). The auxiliary equations are:
dx dy dz dw
x x+y y+z -3w'
of which ole integral only, viz. x3w = const., is algebraic, the other two being logarithmic.
Thus the only possible form off is p (x3w), leading to a set of planes.
II. (ii). The auxiliary equations are:
dx dy dz dw
x x+y z -3w'
three integrals of which are:
y/x- log x = const., z/x = const., x3w = const.,
so that the only algebraic form of f is (z/lx, x3w), and the quartic is the degenerate
surface
(z, x)3 W = 0.
II. (iii). The auxiliary equations are:
dx dy dz dw
x y z - 3w'
three integrals of which are:
y/x = const., z/x = const., x3w = const.,
which lead as before to a degenerate surface
(x, y, z)3 w = 0.