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.

XVII. On Qzartic Sunfaces which admit of Integrals of the first kind of
Total Differentials. By ARTHIR BERRY, M.A., Fellow of King's College,
Cambridge.
[Received 15 November 1899.]
CONTENTS.
~ 1. Introduction.
~ 2. Analysis of the fundamental differential equation.
~ 3. Integration of the differential equation, leading to five possible surfaces.
~ 4. Tabular statement of results.
~ 5. Birational transformation of the surfaces into cones.
6. Numerical genus of surfaces which admit of integrals of the first kind.! 7. Geometrical characteristics of the five surfaces.
~ 1. INTRODUCTION.
THE theory of the Abelian integrals associated with an algebraic plane curve can
be generalised in two distinct ways when we pass from a plane curve to a surface in
three dimensions, that is when we are dealing with an algebraic function of two independent variables. Given an algebraic equation, f(x, y, z) = 0, between three non-homogeneous
variables, we may study either double integrals of the type f R (x, y, z) dxdy, where
R is rational, or single integrals of total differentials of the type (Pdx + Qdy), where
P, Q are rational functions of x, y, z, which satisfy in virtue of f= the condition of
integrability
aP Dq
ay a'
Such integrals of total differentials were introduced into mathematical science by
Picard about fifteen years agoo, and have been the subject of several memoirs by
himl. They have also been studied to some extent by Poincaré++, Noether~, Cayley and
others. The most important results hitherto obtained are given in the "Théorie des
* Comptes Rendus, t. 99 (1 Dec. 1884). drticke," Math. Ann. t. 29 (1887).
t The most important appeared in Liouville, ser. iv. il Note sur le mémoire de M. Picard "Sur les intégrales
t. 1 (1885), and ser. Iv.. 5 (1889). There have also been de différentielles totales algébriques de première espèce,"
a series of notes in the Comptes Rendus. B'ull. des Sciences lMath. ser. II. t. x. (1886): Coll. iMath.
+ Comptes Rendus, t. 99 (29 Dec. 1884). Pcpers, t. xII. no. 852.
~ "Ueber die totalen algebraisehen Differentialaus