XVII. On Quartic Surfaces which admit of Integrals of the first kind of Total Differentials. By ARTHUR BERRY, M.A., Fellow of King's College, Cambridge.

§ 3.

§ 4.

Integration of the differential equation, leading to five possible surfaces.
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, ƒ(x, y, z) = 0, between three non-homogeneous variables, we may study either double integrals of the type R (x, y, z) dady, 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=0 the condition of integrability

Such integrals of total differentials were introduced into mathematical science by Picard about fifteen years ago*, and have been the subject of several memoirs by him†. They have also been studied to some extent by Poincarét, Noethers, Cayley|| and others. The most important results hitherto obtained are given in the "Théorie des

Comptes Rendus, t. 99 (1 Dec. 1884).

The most important appeared in Liouville, ser. IV. t. 1 (1885), and ser. iv. t. 5 (1889). There have also been

a series of notes in the Comptes Rendus.

66

Comptes Rendus, t. 99 (29 Dec. 1884).

§ Ueber die totalen algebraischen Differentialaus

drücke," Math. Ann. t. 29 (1887).

Note sur le mémoire de M. Picard "Sur les intégrales de différentielles totales algébriques de première espèce," Bull. des Sciences Math. ser. II. t. x. (1886): Coll. Math. Papers, t. XII. no. 852.

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 may 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 problem 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 Poincarét, 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 Picard 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 ƒ satisfies the partial differential equation

where 01, 0, 0, 0, are quantics of order n 3, which satisfy the equation

satisfies the condition of integrability, and its integral is finite everywhere with the possible exception of certain singular points and lines.

When n=4 the quantics are linear and the equation (1) becomes a familiar partial differential equation.

If we write

O1 = a;x+by+c;z+d;w, (i=1, 2, 3, 4),

then, in accordance with the usual elementary theory, the integration of the equation (1) depends upon the roots of the algebraic equation

and deduce the general integral of the partial differential equation. But if two or more roots of A=0 are equal the integration of the system (4) is less simple, and one or more of the integrals is in general logarithmic, though these integrals may again become algebraic if the coefficients of the O's satisfy certain further conditions. Although the complete discussion of these cases by quite elementary methods presents no serious difficulty it is rather long and tedious, and the work can be considerably abbreviated by reducing the equations (4) to a standard form by means of the method which was given by Weierstrass as an application of his theory of bilinear forms*. This method, stated in a form applicable to our particular problem, depends on the resolution of the determinant ▲ into "elementary factors" (Elementartheiler). If s a occurs p-tuply as

a factor of A, p-tuply as a factor of each first minor of A, p-tuply as a factor of each

"Bemerkungen zur Integration eines Systems linearer Differentialgleichungen mit constanten Coefficienten," Mathematische Werke, II. pp. 75—6.

These factors are shewn by Weierstrass to be invariant for linear transformation of the variables, and the system of differential equations

is shewn to to be reducible by linear transformation of the dependent variables to a standard form, in which there are as many distinct sets of equations as there are elementary divisors of A, the set corresponding to an elementary divisor (a) being of the form

Applying this theory to our equation we see that the possible ways in which A can be resolved into elementary factors are as follows:

(V) All the roots of ▲ = 0 distinct: (sa), (s — b), (s — c), (s — d).

Also the equation (2) shews that the sum of the roots of A=0 vanishes, so that we must have in

CASE II. b=-3a0, and we may evidently take a = 1, b = −3,

It follows at once that the Case I. (v) is impossible.

For the purposes of our problem we do not want the general integral of the equation (1), but only such integrals as are homogeneous quartics; we may also leave cones out of account, and we must reject solutions giving degenerate (reducible) quartic surfaces; we find also that in one or two other cases we arrive at surfaces which are obviously rational and must therefore be rejected.

§ 3. INTEGRATION OF THE DIFFERENTIAL EQUATION, LEADING TO FIVE POSSIBLE SURFACES.

We have in all (after rejecting I. (v)) thirteen cases to consider, which will now be dealt with seriatim. In each case the transformed variables will still be denoted by x, y, z, w, and the auxiliary equations will be expressed in the usual Lagrangean form, the variable t used by Weierstrass being omitted.

The only quartic of this form is a sum of terms

x1, x2 (y2 — 2zx), (y2 – 2zx)2, x (y3 + 3x2w — 3xyz),

so that w occurs linearly or not at all, and the surface is therefore a cone or rational.