Colloquium publications.

LINEAR SYSTEMS OF CURVES.

29

then as the functions ( and r are assumed to remain finite
throughout the surface f 0, and are seen by the foregoing to be
rational in x, y, z, they can be nothing but constants, as was to be
proven. The double integral of first kind on the surface is
ffdmu dv; the proof that it is unique is closely similar to the
above. Functions of the properties required for F1, F2, F3,
are readily expressed by quotients of theta-functions of two
variables.
Surfaces of this sort are called by Picard and Humbert hyperelliptic surfaces. They are to be distinguished carefully from surfaces whose plane sections are hyperelliptic, or which have a linear
net of hyperelliptic curves upon them, for those we have seen to
be rational (p, = 0); while these, possessing one double integral
everywhere finite, have pg = 1.
Hyperelliptic surfaces of order lower than the sixth do not
exist, as was said. This evokes recollections of Kummer's surface of the fourth order; but that, as Picard shows, is not of this
class, because it has two sets of values (u, v) for every point (x, y, z).
Humbert has discussed hyperelliptic surfaces in extenso,* in particular those of sixth order. He extends this mode of establishing
their existence by theta-formulae, so as to employ the next higher
class of thetas, those in three independent variables. In this way he
reaches surfaces containing three linearly independent exact linear
differentials of the first kind and proves that their order must be
higher than six. An example is given of the eighth order, but the
order seven is left in doubt. Of such representation of these surfaces, the chief advantage is that every algebraic curve lying in the
surface is given by the vanishing of some theta function, so that by
the use of theorems more or less familiar in the theory of thetas,
one obtains an exhaustive treatment of geometry upon a surface.
It is apparent that this line of investigation opens a prospect
of a classification of surfaces based on properties much more general than those merely projective. As was indicated in a remark
upon quartics, this calls for the projective study (for the sake of
* Lioville, ser. 4, vol. 5 (1889), vol. 9 (1893), and ser. 5, vol. 2 (1896).