Colloquium publications.

LINEAR SYSTEMS OF CURVES.

27

covariant of the forms 01, 02, 03, 04 occurring in the integral. But
the surface may be not determinate. (In the above example it
had still 8 free parameters.) Also the O's depend on the choice
of planes of reference. Hence more precisely one should seek to
determine the mixed form f(x, u) (connex) covariant with the
connex
0 = 101 + u202 + U303 + it404
such that every set of values (iu) makes the surface f(x, u) = 0 a
surface possessing the integral of the first kind represented in (11).
In other words the connex 0 is to satisfy the relation
20 020 D20 a20
while the covariant f(x, u), or f, is to be of order in the (x) higher
by 3 than 0, and shall satisfy also identically the equation:
a0 af +o af a0O aO af o.
aul xI a U x2 x u2 -u3 x 3 au cax4 -
Of course the chief interest in this problem would be found in
the lower orders, 4, 5, 6. It might be possible to solve a similar
problem of the theory of forms when the surface is to have two or
more independent integrals of the first kind.
To return to surfaces with two independent exact differentials
of the first kind, we note two theorems of Picard. The existence
of two such differentials is impossible upon any surface of order
m < 6. If a surface have two such differentials, its plane sections
are curves of deficiency at least p = 2, and its geometrical deficiency
is pg 1.
Picard establishes directly the existence of a class of surfaces
with two differentials, in brief as follows: Let the Cartesian
coordinates of a point be given as uniform functions, quadruply
periodic, of two independent variables. Let the relation be such
that to every point (x, y, z) of the surface there corresponds one
and only one pair of values of the two independent variables