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.

334 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
Fonctions Algébriques de deux variables indépendantes" recently (1897) published by
Picard and Simart, a book to which it will in general be convenient to refer.
Integrals of total differentials, like ordinary Abelian integrals, fall into three classes,
of which the first consists of integrals which are always finite. But whereas the
number of linearly independent integrals of the first kind associated with a plane curve
is at once expressible by a simple formula in terms of the singularities of the curve,
and such integrals always exist if the curve has less than its maximum number of
singularities, the corresponding problem for integrals of total differentials is far less
simple and has only been solved for special classes of surfaces. On a cone, an integral
of a total differential is equivalent to an Abelian integral on a plane section of the
cone, so that no new problem arises. Moreover, according to Cayley*, any ruled surface
niay be birationally transformed into a cone, the genus (deficiency) of a section of which
is equal to that of a general plane section of the original surface; hence the number of
integrals of the first kind on a ruled surface can at once be determined, but I am not
aware that there is any known process whereby the transformation can in general be
effected or the integrals actually constructed. For other classes of surfaces the most
important results so far obtained are negative in character; thus it is evident that no
integrals of the first kind can exist on a rational (unicursal) surface, and the same
proposition has been established for surfaces without any singular points or singular lines.
The determination of surfaces or classes of surfaces which admit integrals of the
first kind of total differentials appears therefore to be a problern of some interest.
Since quadrics and cubic surfaces (other than non-singular cones) are rational, they
can possess no integrals of the first kind. Two non-conical quartics possessing such
integrals were discovered by Poincaré+, and stated to be the only possible ones.
Poincaré's results have been adopted by Picard, who has given a proof in outline.
The object of this paper is to establish the existence of certain other quartic
surfaces which have the property in question, but have apparently been overlooked by
the two eminent mathematicians just named. The method which I have adopted appears
to shew also that the list given is complete.
2. ANALYSIS OF THE FUNDAMENTAL DIFFERENTIAL EQUATION.
It has been shewn by Picardl| that if a surface of order n, of which the equation
in homogeneous point coordinates is f(x, y, z, w)=0, admits of an integral of the first
kind, then f satisfies the partial differential equation
01 + 0~ f~ + 03 04 o............................(1),
'6w Dy az aw
* " On the deficiency of certain surfaces," Math. Ann. + Comptes Rendus, t. 99 (29 Dec. 1884).
t. II. (1871): Coll. Math. Papers, t. vIII. no. 524. ~ Picard et Simrart, pp. 135, 136.
+ Picard et Simart, pp. 113, 119, 120. l Ib., Chapter V.